\(\int \frac {1}{(b d+2 c d x)^{7/2} (a+b x+c x^2)^3} \, dx\) [1319]

Optimal result

Integrand size = 26, antiderivative size = 256 \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^3} \, dx=\frac {234 c^2}{5 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2}}+\frac {234 c^2}{\left (b^2-4 a c\right )^4 d^3 \sqrt {b d+2 c d x}}-\frac {1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}+\frac {13 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}+\frac {117 c^2 \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{17/4} d^{7/2}}-\frac {117 c^2 \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{17/4} d^{7/2}} \]

[Out]

234/5*c^2/(-4*a*c+b^2)^3/d/(2*c*d*x+b*d)^(5/2)-1/2/(-4*a*c+b^2)/d/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^2+13/2*c/(
-4*a*c+b^2)^2/d/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)+117*c^2*arctan((d*(2*c*x+b))^(1/2)/(-4*a*c+b^2)^(1/4)/d^(1/2
))/(-4*a*c+b^2)^(17/4)/d^(7/2)-117*c^2*arctanh((d*(2*c*x+b))^(1/2)/(-4*a*c+b^2)^(1/4)/d^(1/2))/(-4*a*c+b^2)^(1
7/4)/d^(7/2)+234*c^2/(-4*a*c+b^2)^4/d^3/(2*c*d*x+b*d)^(1/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {701, 707, 708, 335, 304, 209, 212} \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^3} \, dx=\frac {117 c^2 \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{17/4}}-\frac {117 c^2 \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{17/4}}+\frac {234 c^2}{d^3 \left (b^2-4 a c\right )^4 \sqrt {b d+2 c d x}}+\frac {234 c^2}{5 d \left (b^2-4 a c\right )^3 (b d+2 c d x)^{5/2}}+\frac {13 c}{2 d \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) (b d+2 c d x)^{5/2}}-\frac {1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 (b d+2 c d x)^{5/2}} \]

[In]

Int[1/((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^3),x]

[Out]

(234*c^2)/(5*(b^2 - 4*a*c)^3*d*(b*d + 2*c*d*x)^(5/2)) + (234*c^2)/((b^2 - 4*a*c)^4*d^3*Sqrt[b*d + 2*c*d*x]) -
1/(2*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^2) + (13*c)/(2*(b^2 - 4*a*c)^2*d*(b*d + 2*c*d*x)^
(5/2)*(a + b*x + c*x^2)) + (117*c^2*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^
(17/4)*d^(7/2)) - (117*c^2*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^(17/4)*d
^(7/2))

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 701

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2*c*(d + e*x)^(m + 1
)*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^2 - 4*a*c))), x] - Dist[2*c*e*((m + 2*p + 3)/(e*(p + 1)*(b^2 - 4*a*
c))), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0]
&& EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[p, -1] &&  !GtQ[m, 1] && RationalQ[m] && IntegerQ[2*p]

Rule 707

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[-2*b*d*(d + e*x)^(m
+ 1)*((a + b*x + c*x^2)^(p + 1)/(d^2*(m + 1)*(b^2 - 4*a*c))), x] + Dist[b^2*((m + 2*p + 3)/(d^2*(m + 1)*(b^2 -
4*a*c))), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*
c, 0] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[m, -1] && (IntegerQ[2*p] || (IntegerQ[m] && Rationa
lQ[p]) || IntegerQ[(m + 2*p + 3)/2])

Rule 708

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[x^m*(
a - b^2/(4*c) + (c*x^2)/e^2)^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0]
&& EqQ[2*c*d - b*e, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}-\frac {(13 c) \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )} \\ & = -\frac {1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}+\frac {13 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}+\frac {\left (117 c^2\right ) \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )} \, dx}{2 \left (b^2-4 a c\right )^2} \\ & = \frac {234 c^2}{5 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2}}-\frac {1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}+\frac {13 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}+\frac {\left (117 c^2\right ) \int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )} \, dx}{2 \left (b^2-4 a c\right )^3 d^2} \\ & = \frac {234 c^2}{5 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2}}+\frac {234 c^2}{\left (b^2-4 a c\right )^4 d^3 \sqrt {b d+2 c d x}}-\frac {1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}+\frac {13 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}+\frac {\left (117 c^2\right ) \int \frac {\sqrt {b d+2 c d x}}{a+b x+c x^2} \, dx}{2 \left (b^2-4 a c\right )^4 d^4} \\ & = \frac {234 c^2}{5 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2}}+\frac {234 c^2}{\left (b^2-4 a c\right )^4 d^3 \sqrt {b d+2 c d x}}-\frac {1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}+\frac {13 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}+\frac {(117 c) \text {Subst}\left (\int \frac {\sqrt {x}}{a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}} \, dx,x,b d+2 c d x\right )}{4 \left (b^2-4 a c\right )^4 d^5} \\ & = \frac {234 c^2}{5 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2}}+\frac {234 c^2}{\left (b^2-4 a c\right )^4 d^3 \sqrt {b d+2 c d x}}-\frac {1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}+\frac {13 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}+\frac {(117 c) \text {Subst}\left (\int \frac {x^2}{a-\frac {b^2}{4 c}+\frac {x^4}{4 c d^2}} \, dx,x,\sqrt {d (b+2 c x)}\right )}{2 \left (b^2-4 a c\right )^4 d^5} \\ & = \frac {234 c^2}{5 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2}}+\frac {234 c^2}{\left (b^2-4 a c\right )^4 d^3 \sqrt {b d+2 c d x}}-\frac {1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}+\frac {13 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}-\frac {\left (117 c^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c} d-x^2} \, dx,x,\sqrt {d (b+2 c x)}\right )}{\left (b^2-4 a c\right )^4 d^3}+\frac {\left (117 c^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c} d+x^2} \, dx,x,\sqrt {d (b+2 c x)}\right )}{\left (b^2-4 a c\right )^4 d^3} \\ & = \frac {234 c^2}{5 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2}}+\frac {234 c^2}{\left (b^2-4 a c\right )^4 d^3 \sqrt {b d+2 c d x}}-\frac {1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}+\frac {13 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}+\frac {117 c^2 \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{17/4} d^{7/2}}-\frac {117 c^2 \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{17/4} d^{7/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.63 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.39 \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^3} \, dx=\frac {\left (\frac {1}{10}+\frac {i}{10}\right ) c^2 \left (-\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) (b+2 c x) \left (-32 b^6+384 a b^4 c-1536 a^2 b^2 c^2+2048 a^3 c^3-416 b^4 (b+2 c x)^2+3328 a b^2 c (b+2 c x)^2-6656 a^2 c^2 (b+2 c x)^2+1053 b^2 (b+2 c x)^4-4212 a c (b+2 c x)^4-585 (b+2 c x)^6\right )}{c^2 \left (b^2-4 a c\right )^4 (a+x (b+c x))^2}-\frac {585 (b+2 c x)^{7/2} \arctan \left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{17/4}}+\frac {585 (b+2 c x)^{7/2} \arctan \left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{17/4}}-\frac {585 (b+2 c x)^{7/2} \text {arctanh}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )}{\left (b^2-4 a c\right )^{17/4}}\right )}{(d (b+2 c x))^{7/2}} \]

[In]

Integrate[1/((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^3),x]

[Out]

((1/10 + I/10)*c^2*(((-1/8 + I/8)*(b + 2*c*x)*(-32*b^6 + 384*a*b^4*c - 1536*a^2*b^2*c^2 + 2048*a^3*c^3 - 416*b
^4*(b + 2*c*x)^2 + 3328*a*b^2*c*(b + 2*c*x)^2 - 6656*a^2*c^2*(b + 2*c*x)^2 + 1053*b^2*(b + 2*c*x)^4 - 4212*a*c
*(b + 2*c*x)^4 - 585*(b + 2*c*x)^6))/(c^2*(b^2 - 4*a*c)^4*(a + x*(b + c*x))^2) - (585*(b + 2*c*x)^(7/2)*ArcTan
[1 - ((1 + I)*Sqrt[b + 2*c*x])/(b^2 - 4*a*c)^(1/4)])/(b^2 - 4*a*c)^(17/4) + (585*(b + 2*c*x)^(7/2)*ArcTan[1 +
((1 + I)*Sqrt[b + 2*c*x])/(b^2 - 4*a*c)^(1/4)])/(b^2 - 4*a*c)^(17/4) - (585*(b + 2*c*x)^(7/2)*ArcTanh[((1 + I)
*(b^2 - 4*a*c)^(1/4)*Sqrt[b + 2*c*x])/(Sqrt[b^2 - 4*a*c] + I*(b + 2*c*x))])/(b^2 - 4*a*c)^(17/4)))/(d*(b + 2*c
*x))^(7/2)

Maple [A] (verified)

Time = 3.11 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.49

method	result	size
derivativedivides	\(64 c^{2} d^{5} \left (-\frac {1}{5 d^{6} \left (4 a c -b^{2}\right )^{3} \left (2 c d x +b d \right )^{\frac {5}{2}}}+\frac {3}{d^{8} \left (4 a c -b^{2}\right )^{4} \sqrt {2 c d x +b d}}+\frac {\frac {\frac {21 \left (2 c d x +b d \right )^{\frac {7}{2}}}{32}+16 \left (\frac {25}{128} a c \,d^{2}-\frac {25}{512} b^{2} d^{2}\right ) \left (2 c d x +b d \right )^{\frac {3}{2}}}{\left (\left (2 c d x +b d \right )^{2}+4 a c \,d^{2}-b^{2} d^{2}\right )^{2}}+\frac {117 \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{256 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}}{d^{8} \left (4 a c -b^{2}\right )^{4}}\right )\)	\(382\)
default	\(64 c^{2} d^{5} \left (-\frac {1}{5 d^{6} \left (4 a c -b^{2}\right )^{3} \left (2 c d x +b d \right )^{\frac {5}{2}}}+\frac {3}{d^{8} \left (4 a c -b^{2}\right )^{4} \sqrt {2 c d x +b d}}+\frac {\frac {\frac {21 \left (2 c d x +b d \right )^{\frac {7}{2}}}{32}+16 \left (\frac {25}{128} a c \,d^{2}-\frac {25}{512} b^{2} d^{2}\right ) \left (2 c d x +b d \right )^{\frac {3}{2}}}{\left (\left (2 c d x +b d \right )^{2}+4 a c \,d^{2}-b^{2} d^{2}\right )^{2}}+\frac {117 \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{256 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}}{d^{8} \left (4 a c -b^{2}\right )^{4}}\right )\)	\(382\)
pseudoelliptic	\(-\frac {-\frac {585 c^{2} \sqrt {2}\, \left (2 c x +b \right )^{2} \left (c \,x^{2}+b x +a \right )^{2} \left (\ln \left (\frac {\sqrt {d^{2} \left (4 a c -b^{2}\right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+d \left (2 c x +b \right )}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+\sqrt {d^{2} \left (4 a c -b^{2}\right )}+d \left (2 c x +b \right )}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right )\right ) \sqrt {d \left (2 c x +b \right )}}{1024}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \left (-\frac {585 c^{6} x^{6}}{32}-\frac {1053 x^{4} \left (\frac {5 b x}{3}+a \right ) c^{5}}{32}-13 x^{2} \left (\frac {297}{64} b^{2} x^{2}+\frac {81}{16} a b x +a^{2}\right ) c^{4}+\left (a^{3}-\frac {2743}{64} a \,b^{2} x^{2}-13 a^{2} b x -\frac {117}{4} b^{3} x^{3}\right ) c^{3}-4 \left (\frac {13 b x}{8}+a \right ) b^{2} \left (\frac {221 b x}{256}+a \right ) c^{2}-\frac {125 b^{4} \left (\frac {13 b x}{25}+a \right ) c}{512}+\frac {5 b^{6}}{512}\right )}{5 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, d^{3} \left (2 c x +b \right )^{2} \left (c \,x^{2}+b x +a \right )^{2} \left (-\frac {b^{2}}{4}+a c \right )^{4}}\)	\(456\)

[In]

int(1/(2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

64*c^2*d^5*(-1/5/d^6/(4*a*c-b^2)^3/(2*c*d*x+b*d)^(5/2)+3/d^8/(4*a*c-b^2)^4/(2*c*d*x+b*d)^(1/2)+1/d^8/(4*a*c-b^
2)^4*(16*(21/512*(2*c*d*x+b*d)^(7/2)+(25/128*a*c*d^2-25/512*b^2*d^2)*(2*c*d*x+b*d)^(3/2))/((2*c*d*x+b*d)^2+4*a
*c*d^2-b^2*d^2)^2+117/256/(4*a*c*d^2-b^2*d^2)^(1/4)*2^(1/2)*(ln((2*c*d*x+b*d-(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*
x+b*d)^(1/2)*2^(1/2)+(4*a*c*d^2-b^2*d^2)^(1/2))/(2*c*d*x+b*d+(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)*2^(
1/2)+(4*a*c*d^2-b^2*d^2)^(1/2)))+2*arctan(2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1)-2*arctan(-2
^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1))))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.37 (sec) , antiderivative size = 4956, normalized size of antiderivative = 19.36 \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(1/(2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

-1/10*(585*(8*(b^8*c^5 - 16*a*b^6*c^6 + 96*a^2*b^4*c^7 - 256*a^3*b^2*c^8 + 256*a^4*c^9)*d^4*x^7 + 28*(b^9*c^4
- 16*a*b^7*c^5 + 96*a^2*b^5*c^6 - 256*a^3*b^3*c^7 + 256*a^4*b*c^8)*d^4*x^6 + 2*(19*b^10*c^3 - 296*a*b^8*c^4 +
1696*a^2*b^6*c^5 - 4096*a^3*b^4*c^6 + 2816*a^4*b^2*c^7 + 2048*a^5*c^8)*d^4*x^5 + 5*(5*b^11*c^2 - 72*a*b^9*c^3
+ 352*a^2*b^7*c^4 - 512*a^3*b^5*c^5 - 768*a^4*b^3*c^6 + 2048*a^5*b*c^7)*d^4*x^4 + 4*(2*b^12*c - 23*a*b^10*c^2
+ 50*a^2*b^8*c^3 + 320*a^3*b^6*c^4 - 1600*a^4*b^4*c^5 + 1792*a^5*b^2*c^6 + 512*a^6*c^7)*d^4*x^3 + (b^13 - 2*a*
b^11*c - 116*a^2*b^9*c^2 + 896*a^3*b^7*c^3 - 2176*a^4*b^5*c^4 + 512*a^5*b^3*c^5 + 3072*a^6*b*c^6)*d^4*x^2 + 2*
(a*b^12 - 13*a^2*b^10*c + 48*a^3*b^8*c^2 + 32*a^4*b^6*c^3 - 512*a^5*b^4*c^4 + 768*a^6*b^2*c^5)*d^4*x + (a^2*b^
11 - 16*a^3*b^9*c + 96*a^4*b^7*c^2 - 256*a^5*b^5*c^3 + 256*a^6*b^3*c^4)*d^4)*(c^8/((b^34 - 68*a*b^32*c + 2176*
a^2*b^30*c^2 - 43520*a^3*b^28*c^3 + 609280*a^4*b^26*c^4 - 6336512*a^5*b^24*c^5 + 50692096*a^6*b^22*c^6 - 31863
6032*a^7*b^20*c^7 + 1593180160*a^8*b^18*c^8 - 6372720640*a^9*b^16*c^9 + 20392706048*a^10*b^14*c^10 - 519087063
04*a^11*b^12*c^11 + 103817412608*a^12*b^10*c^12 - 159719096320*a^13*b^8*c^13 + 182536110080*a^14*b^6*c^14 - 14
6028888064*a^15*b^4*c^15 + 73014444032*a^16*b^2*c^16 - 17179869184*a^17*c^17)*d^14))^(1/4)*log(1601613*(b^26 -
52*a*b^24*c + 1248*a^2*b^22*c^2 - 18304*a^3*b^20*c^3 + 183040*a^4*b^18*c^4 - 1317888*a^5*b^16*c^5 + 7028736*a
^6*b^14*c^6 - 28114944*a^7*b^12*c^7 + 84344832*a^8*b^10*c^8 - 187432960*a^9*b^8*c^9 + 299892736*a^10*b^6*c^10
- 327155712*a^11*b^4*c^11 + 218103808*a^12*b^2*c^12 - 67108864*a^13*c^13)*d^11*(c^8/((b^34 - 68*a*b^32*c + 217
6*a^2*b^30*c^2 - 43520*a^3*b^28*c^3 + 609280*a^4*b^26*c^4 - 6336512*a^5*b^24*c^5 + 50692096*a^6*b^22*c^6 - 318
636032*a^7*b^20*c^7 + 1593180160*a^8*b^18*c^8 - 6372720640*a^9*b^16*c^9 + 20392706048*a^10*b^14*c^10 - 5190870
6304*a^11*b^12*c^11 + 103817412608*a^12*b^10*c^12 - 159719096320*a^13*b^8*c^13 + 182536110080*a^14*b^6*c^14 -
146028888064*a^15*b^4*c^15 + 73014444032*a^16*b^2*c^16 - 17179869184*a^17*c^17)*d^14))^(3/4) + 1601613*sqrt(2*
c*d*x + b*d)*c^6) + 585*(-8*I*(b^8*c^5 - 16*a*b^6*c^6 + 96*a^2*b^4*c^7 - 256*a^3*b^2*c^8 + 256*a^4*c^9)*d^4*x^
7 - 28*I*(b^9*c^4 - 16*a*b^7*c^5 + 96*a^2*b^5*c^6 - 256*a^3*b^3*c^7 + 256*a^4*b*c^8)*d^4*x^6 - 2*I*(19*b^10*c^
3 - 296*a*b^8*c^4 + 1696*a^2*b^6*c^5 - 4096*a^3*b^4*c^6 + 2816*a^4*b^2*c^7 + 2048*a^5*c^8)*d^4*x^5 - 5*I*(5*b^
11*c^2 - 72*a*b^9*c^3 + 352*a^2*b^7*c^4 - 512*a^3*b^5*c^5 - 768*a^4*b^3*c^6 + 2048*a^5*b*c^7)*d^4*x^4 - 4*I*(2
*b^12*c - 23*a*b^10*c^2 + 50*a^2*b^8*c^3 + 320*a^3*b^6*c^4 - 1600*a^4*b^4*c^5 + 1792*a^5*b^2*c^6 + 512*a^6*c^7
)*d^4*x^3 - I*(b^13 - 2*a*b^11*c - 116*a^2*b^9*c^2 + 896*a^3*b^7*c^3 - 2176*a^4*b^5*c^4 + 512*a^5*b^3*c^5 + 30
72*a^6*b*c^6)*d^4*x^2 - 2*I*(a*b^12 - 13*a^2*b^10*c + 48*a^3*b^8*c^2 + 32*a^4*b^6*c^3 - 512*a^5*b^4*c^4 + 768*
a^6*b^2*c^5)*d^4*x - I*(a^2*b^11 - 16*a^3*b^9*c + 96*a^4*b^7*c^2 - 256*a^5*b^5*c^3 + 256*a^6*b^3*c^4)*d^4)*(c^
8/((b^34 - 68*a*b^32*c + 2176*a^2*b^30*c^2 - 43520*a^3*b^28*c^3 + 609280*a^4*b^26*c^4 - 6336512*a^5*b^24*c^5 +
50692096*a^6*b^22*c^6 - 318636032*a^7*b^20*c^7 + 1593180160*a^8*b^18*c^8 - 6372720640*a^9*b^16*c^9 + 20392706
048*a^10*b^14*c^10 - 51908706304*a^11*b^12*c^11 + 103817412608*a^12*b^10*c^12 - 159719096320*a^13*b^8*c^13 + 1
82536110080*a^14*b^6*c^14 - 146028888064*a^15*b^4*c^15 + 73014444032*a^16*b^2*c^16 - 17179869184*a^17*c^17)*d^
14))^(1/4)*log(1601613*I*(b^26 - 52*a*b^24*c + 1248*a^2*b^22*c^2 - 18304*a^3*b^20*c^3 + 183040*a^4*b^18*c^4 -
1317888*a^5*b^16*c^5 + 7028736*a^6*b^14*c^6 - 28114944*a^7*b^12*c^7 + 84344832*a^8*b^10*c^8 - 187432960*a^9*b^
8*c^9 + 299892736*a^10*b^6*c^10 - 327155712*a^11*b^4*c^11 + 218103808*a^12*b^2*c^12 - 67108864*a^13*c^13)*d^11
*(c^8/((b^34 - 68*a*b^32*c + 2176*a^2*b^30*c^2 - 43520*a^3*b^28*c^3 + 609280*a^4*b^26*c^4 - 6336512*a^5*b^24*c
^5 + 50692096*a^6*b^22*c^6 - 318636032*a^7*b^20*c^7 + 1593180160*a^8*b^18*c^8 - 6372720640*a^9*b^16*c^9 + 2039
2706048*a^10*b^14*c^10 - 51908706304*a^11*b^12*c^11 + 103817412608*a^12*b^10*c^12 - 159719096320*a^13*b^8*c^13
+ 182536110080*a^14*b^6*c^14 - 146028888064*a^15*b^4*c^15 + 73014444032*a^16*b^2*c^16 - 17179869184*a^17*c^17
)*d^14))^(3/4) + 1601613*sqrt(2*c*d*x + b*d)*c^6) + 585*(8*I*(b^8*c^5 - 16*a*b^6*c^6 + 96*a^2*b^4*c^7 - 256*a^
3*b^2*c^8 + 256*a^4*c^9)*d^4*x^7 + 28*I*(b^9*c^4 - 16*a*b^7*c^5 + 96*a^2*b^5*c^6 - 256*a^3*b^3*c^7 + 256*a^4*b
*c^8)*d^4*x^6 + 2*I*(19*b^10*c^3 - 296*a*b^8*c^4 + 1696*a^2*b^6*c^5 - 4096*a^3*b^4*c^6 + 2816*a^4*b^2*c^7 + 20
48*a^5*c^8)*d^4*x^5 + 5*I*(5*b^11*c^2 - 72*a*b^9*c^3 + 352*a^2*b^7*c^4 - 512*a^3*b^5*c^5 - 768*a^4*b^3*c^6 + 2
048*a^5*b*c^7)*d^4*x^4 + 4*I*(2*b^12*c - 23*a*b^10*c^2 + 50*a^2*b^8*c^3 + 320*a^3*b^6*c^4 - 1600*a^4*b^4*c^5 +
1792*a^5*b^2*c^6 + 512*a^6*c^7)*d^4*x^3 + I*(b^13 - 2*a*b^11*c - 116*a^2*b^9*c^2 + 896*a^3*b^7*c^3 - 2176*a^4
*b^5*c^4 + 512*a^5*b^3*c^5 + 3072*a^6*b*c^6)*d^4*x^2 + 2*I*(a*b^12 - 13*a^2*b^10*c + 48*a^3*b^8*c^2 + 32*a^4*b
^6*c^3 - 512*a^5*b^4*c^4 + 768*a^6*b^2*c^5)*d^4*x + I*(a^2*b^11 - 16*a^3*b^9*c + 96*a^4*b^7*c^2 - 256*a^5*b^5*
c^3 + 256*a^6*b^3*c^4)*d^4)*(c^8/((b^34 - 68*a*b^32*c + 2176*a^2*b^30*c^2 - 43520*a^3*b^28*c^3 + 609280*a^4*b^
26*c^4 - 6336512*a^5*b^24*c^5 + 50692096*a^6*b^22*c^6 - 318636032*a^7*b^20*c^7 + 1593180160*a^8*b^18*c^8 - 637
2720640*a^9*b^16*c^9 + 20392706048*a^10*b^14*c^10 - 51908706304*a^11*b^12*c^11 + 103817412608*a^12*b^10*c^12 -
159719096320*a^13*b^8*c^13 + 182536110080*a^14*b^6*c^14 - 146028888064*a^15*b^4*c^15 + 73014444032*a^16*b^2*c
^16 - 17179869184*a^17*c^17)*d^14))^(1/4)*log(-1601613*I*(b^26 - 52*a*b^24*c + 1248*a^2*b^22*c^2 - 18304*a^3*b
^20*c^3 + 183040*a^4*b^18*c^4 - 1317888*a^5*b^16*c^5 + 7028736*a^6*b^14*c^6 - 28114944*a^7*b^12*c^7 + 84344832
*a^8*b^10*c^8 - 187432960*a^9*b^8*c^9 + 299892736*a^10*b^6*c^10 - 327155712*a^11*b^4*c^11 + 218103808*a^12*b^2
*c^12 - 67108864*a^13*c^13)*d^11*(c^8/((b^34 - 68*a*b^32*c + 2176*a^2*b^30*c^2 - 43520*a^3*b^28*c^3 + 609280*a
^4*b^26*c^4 - 6336512*a^5*b^24*c^5 + 50692096*a^6*b^22*c^6 - 318636032*a^7*b^20*c^7 + 1593180160*a^8*b^18*c^8
- 6372720640*a^9*b^16*c^9 + 20392706048*a^10*b^14*c^10 - 51908706304*a^11*b^12*c^11 + 103817412608*a^12*b^10*c
^12 - 159719096320*a^13*b^8*c^13 + 182536110080*a^14*b^6*c^14 - 146028888064*a^15*b^4*c^15 + 73014444032*a^16*
b^2*c^16 - 17179869184*a^17*c^17)*d^14))^(3/4) + 1601613*sqrt(2*c*d*x + b*d)*c^6) - 585*(8*(b^8*c^5 - 16*a*b^6
*c^6 + 96*a^2*b^4*c^7 - 256*a^3*b^2*c^8 + 256*a^4*c^9)*d^4*x^7 + 28*(b^9*c^4 - 16*a*b^7*c^5 + 96*a^2*b^5*c^6 -
256*a^3*b^3*c^7 + 256*a^4*b*c^8)*d^4*x^6 + 2*(19*b^10*c^3 - 296*a*b^8*c^4 + 1696*a^2*b^6*c^5 - 4096*a^3*b^4*c
^6 + 2816*a^4*b^2*c^7 + 2048*a^5*c^8)*d^4*x^5 + 5*(5*b^11*c^2 - 72*a*b^9*c^3 + 352*a^2*b^7*c^4 - 512*a^3*b^5*c
^5 - 768*a^4*b^3*c^6 + 2048*a^5*b*c^7)*d^4*x^4 + 4*(2*b^12*c - 23*a*b^10*c^2 + 50*a^2*b^8*c^3 + 320*a^3*b^6*c^
4 - 1600*a^4*b^4*c^5 + 1792*a^5*b^2*c^6 + 512*a^6*c^7)*d^4*x^3 + (b^13 - 2*a*b^11*c - 116*a^2*b^9*c^2 + 896*a^
3*b^7*c^3 - 2176*a^4*b^5*c^4 + 512*a^5*b^3*c^5 + 3072*a^6*b*c^6)*d^4*x^2 + 2*(a*b^12 - 13*a^2*b^10*c + 48*a^3*
b^8*c^2 + 32*a^4*b^6*c^3 - 512*a^5*b^4*c^4 + 768*a^6*b^2*c^5)*d^4*x + (a^2*b^11 - 16*a^3*b^9*c + 96*a^4*b^7*c^
2 - 256*a^5*b^5*c^3 + 256*a^6*b^3*c^4)*d^4)*(c^8/((b^34 - 68*a*b^32*c + 2176*a^2*b^30*c^2 - 43520*a^3*b^28*c^3
+ 609280*a^4*b^26*c^4 - 6336512*a^5*b^24*c^5 + 50692096*a^6*b^22*c^6 - 318636032*a^7*b^20*c^7 + 1593180160*a^
8*b^18*c^8 - 6372720640*a^9*b^16*c^9 + 20392706048*a^10*b^14*c^10 - 51908706304*a^11*b^12*c^11 + 103817412608*
a^12*b^10*c^12 - 159719096320*a^13*b^8*c^13 + 182536110080*a^14*b^6*c^14 - 146028888064*a^15*b^4*c^15 + 730144
44032*a^16*b^2*c^16 - 17179869184*a^17*c^17)*d^14))^(1/4)*log(-1601613*(b^26 - 52*a*b^24*c + 1248*a^2*b^22*c^2
- 18304*a^3*b^20*c^3 + 183040*a^4*b^18*c^4 - 1317888*a^5*b^16*c^5 + 7028736*a^6*b^14*c^6 - 28114944*a^7*b^12*
c^7 + 84344832*a^8*b^10*c^8 - 187432960*a^9*b^8*c^9 + 299892736*a^10*b^6*c^10 - 327155712*a^11*b^4*c^11 + 2181
03808*a^12*b^2*c^12 - 67108864*a^13*c^13)*d^11*(c^8/((b^34 - 68*a*b^32*c + 2176*a^2*b^30*c^2 - 43520*a^3*b^28*
c^3 + 609280*a^4*b^26*c^4 - 6336512*a^5*b^24*c^5 + 50692096*a^6*b^22*c^6 - 318636032*a^7*b^20*c^7 + 1593180160
*a^8*b^18*c^8 - 6372720640*a^9*b^16*c^9 + 20392706048*a^10*b^14*c^10 - 51908706304*a^11*b^12*c^11 + 1038174126
08*a^12*b^10*c^12 - 159719096320*a^13*b^8*c^13 + 182536110080*a^14*b^6*c^14 - 146028888064*a^15*b^4*c^15 + 730
14444032*a^16*b^2*c^16 - 17179869184*a^17*c^17)*d^14))^(3/4) + 1601613*sqrt(2*c*d*x + b*d)*c^6) - (9360*c^6*x^
6 + 28080*b*c^5*x^5 - 5*b^6 + 125*a*b^4*c + 2048*a^2*b^2*c^2 - 512*a^3*c^3 + 2808*(11*b^2*c^4 + 6*a*c^5)*x^4 +
3744*(4*b^3*c^3 + 9*a*b*c^4)*x^3 + 13*(221*b^4*c^2 + 1688*a*b^2*c^3 + 512*a^2*c^4)*x^2 + 13*(5*b^5*c + 392*a*
b^3*c^2 + 512*a^2*b*c^3)*x)*sqrt(2*c*d*x + b*d))/(8*(b^8*c^5 - 16*a*b^6*c^6 + 96*a^2*b^4*c^7 - 256*a^3*b^2*c^8
+ 256*a^4*c^9)*d^4*x^7 + 28*(b^9*c^4 - 16*a*b^7*c^5 + 96*a^2*b^5*c^6 - 256*a^3*b^3*c^7 + 256*a^4*b*c^8)*d^4*x
^6 + 2*(19*b^10*c^3 - 296*a*b^8*c^4 + 1696*a^2*b^6*c^5 - 4096*a^3*b^4*c^6 + 2816*a^4*b^2*c^7 + 2048*a^5*c^8)*d
^4*x^5 + 5*(5*b^11*c^2 - 72*a*b^9*c^3 + 352*a^2*b^7*c^4 - 512*a^3*b^5*c^5 - 768*a^4*b^3*c^6 + 2048*a^5*b*c^7)*
d^4*x^4 + 4*(2*b^12*c - 23*a*b^10*c^2 + 50*a^2*b^8*c^3 + 320*a^3*b^6*c^4 - 1600*a^4*b^4*c^5 + 1792*a^5*b^2*c^6
+ 512*a^6*c^7)*d^4*x^3 + (b^13 - 2*a*b^11*c - 116*a^2*b^9*c^2 + 896*a^3*b^7*c^3 - 2176*a^4*b^5*c^4 + 512*a^5*
b^3*c^5 + 3072*a^6*b*c^6)*d^4*x^2 + 2*(a*b^12 - 13*a^2*b^10*c + 48*a^3*b^8*c^2 + 32*a^4*b^6*c^3 - 512*a^5*b^4*
c^4 + 768*a^6*b^2*c^5)*d^4*x + (a^2*b^11 - 16*a^3*b^9*c + 96*a^4*b^7*c^2 - 256*a^5*b^5*c^3 + 256*a^6*b^3*c^4)*
d^4)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(1/(2*c*d*x+b*d)**(7/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 958 vs. \(2 (222) = 444\).

Time = 0.34 (sec) , antiderivative size = 958, normalized size of antiderivative = 3.74 \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^3} \, dx=-\frac {117 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} + 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right )}{\sqrt {2} b^{10} d^{5} - 20 \, \sqrt {2} a b^{8} c d^{5} + 160 \, \sqrt {2} a^{2} b^{6} c^{2} d^{5} - 640 \, \sqrt {2} a^{3} b^{4} c^{3} d^{5} + 1280 \, \sqrt {2} a^{4} b^{2} c^{4} d^{5} - 1024 \, \sqrt {2} a^{5} c^{5} d^{5}} - \frac {117 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} - 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right )}{\sqrt {2} b^{10} d^{5} - 20 \, \sqrt {2} a b^{8} c d^{5} + 160 \, \sqrt {2} a^{2} b^{6} c^{2} d^{5} - 640 \, \sqrt {2} a^{3} b^{4} c^{3} d^{5} + 1280 \, \sqrt {2} a^{4} b^{2} c^{4} d^{5} - 1024 \, \sqrt {2} a^{5} c^{5} d^{5}} + \frac {117 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c^{2} \log \left (2 \, c d x + b d + \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \, {\left (\sqrt {2} b^{10} d^{5} - 20 \, \sqrt {2} a b^{8} c d^{5} + 160 \, \sqrt {2} a^{2} b^{6} c^{2} d^{5} - 640 \, \sqrt {2} a^{3} b^{4} c^{3} d^{5} + 1280 \, \sqrt {2} a^{4} b^{2} c^{4} d^{5} - 1024 \, \sqrt {2} a^{5} c^{5} d^{5}\right )}} - \frac {117 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c^{2} \log \left (2 \, c d x + b d - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \, {\left (\sqrt {2} b^{10} d^{5} - 20 \, \sqrt {2} a b^{8} c d^{5} + 160 \, \sqrt {2} a^{2} b^{6} c^{2} d^{5} - 640 \, \sqrt {2} a^{3} b^{4} c^{3} d^{5} + 1280 \, \sqrt {2} a^{4} b^{2} c^{4} d^{5} - 1024 \, \sqrt {2} a^{5} c^{5} d^{5}\right )}} - \frac {2 \, {\left (25 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} c^{2} d^{2} - 100 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} a c^{3} d^{2} - 21 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} c^{2}\right )}}{{\left (b^{8} d^{3} - 16 \, a b^{6} c d^{3} + 96 \, a^{2} b^{4} c^{2} d^{3} - 256 \, a^{3} b^{2} c^{3} d^{3} + 256 \, a^{4} c^{4} d^{3}\right )} {\left (b^{2} d^{2} - 4 \, a c d^{2} - {\left (2 \, c d x + b d\right )}^{2}\right )}^{2}} + \frac {64 \, {\left (b^{2} c^{2} d^{2} - 4 \, a c^{3} d^{2} + 15 \, {\left (2 \, c d x + b d\right )}^{2} c^{2}\right )}}{5 \, {\left (b^{8} d^{3} - 16 \, a b^{6} c d^{3} + 96 \, a^{2} b^{4} c^{2} d^{3} - 256 \, a^{3} b^{2} c^{3} d^{3} + 256 \, a^{4} c^{4} d^{3}\right )} {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}} \]

[In]

integrate(1/(2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

-117*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c^2*arctan(1/2*sqrt(2)*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4) + 2*sqrt(2*c*d*
x + b*d))/(-b^2*d^2 + 4*a*c*d^2)^(1/4))/(sqrt(2)*b^10*d^5 - 20*sqrt(2)*a*b^8*c*d^5 + 160*sqrt(2)*a^2*b^6*c^2*d
^5 - 640*sqrt(2)*a^3*b^4*c^3*d^5 + 1280*sqrt(2)*a^4*b^2*c^4*d^5 - 1024*sqrt(2)*a^5*c^5*d^5) - 117*(-b^2*d^2 +
4*a*c*d^2)^(3/4)*c^2*arctan(-1/2*sqrt(2)*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4) - 2*sqrt(2*c*d*x + b*d))/(-b^2*
d^2 + 4*a*c*d^2)^(1/4))/(sqrt(2)*b^10*d^5 - 20*sqrt(2)*a*b^8*c*d^5 + 160*sqrt(2)*a^2*b^6*c^2*d^5 - 640*sqrt(2)
*a^3*b^4*c^3*d^5 + 1280*sqrt(2)*a^4*b^2*c^4*d^5 - 1024*sqrt(2)*a^5*c^5*d^5) + 117/2*(-b^2*d^2 + 4*a*c*d^2)^(3/
4)*c^2*log(2*c*d*x + b*d + sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*sqrt(2*c*d*x + b*d) + sqrt(-b^2*d^2 + 4*a*c*d^
2))/(sqrt(2)*b^10*d^5 - 20*sqrt(2)*a*b^8*c*d^5 + 160*sqrt(2)*a^2*b^6*c^2*d^5 - 640*sqrt(2)*a^3*b^4*c^3*d^5 + 1
280*sqrt(2)*a^4*b^2*c^4*d^5 - 1024*sqrt(2)*a^5*c^5*d^5) - 117/2*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c^2*log(2*c*d*x +
b*d - sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*sqrt(2*c*d*x + b*d) + sqrt(-b^2*d^2 + 4*a*c*d^2))/(sqrt(2)*b^10*d^
5 - 20*sqrt(2)*a*b^8*c*d^5 + 160*sqrt(2)*a^2*b^6*c^2*d^5 - 640*sqrt(2)*a^3*b^4*c^3*d^5 + 1280*sqrt(2)*a^4*b^2*
c^4*d^5 - 1024*sqrt(2)*a^5*c^5*d^5) - 2*(25*(2*c*d*x + b*d)^(3/2)*b^2*c^2*d^2 - 100*(2*c*d*x + b*d)^(3/2)*a*c^
3*d^2 - 21*(2*c*d*x + b*d)^(7/2)*c^2)/((b^8*d^3 - 16*a*b^6*c*d^3 + 96*a^2*b^4*c^2*d^3 - 256*a^3*b^2*c^3*d^3 +
256*a^4*c^4*d^3)*(b^2*d^2 - 4*a*c*d^2 - (2*c*d*x + b*d)^2)^2) + 64/5*(b^2*c^2*d^2 - 4*a*c^3*d^2 + 15*(2*c*d*x
+ b*d)^2*c^2)/((b^8*d^3 - 16*a*b^6*c*d^3 + 96*a^2*b^4*c^2*d^3 - 256*a^3*b^2*c^3*d^3 + 256*a^4*c^4*d^3)*(2*c*d*
x + b*d)^(5/2))

Mupad [B] (verification not implemented)

Time = 9.86 (sec) , antiderivative size = 543, normalized size of antiderivative = 2.12 \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^3} \, dx=\frac {\frac {234\,c^2\,{\left (b\,d+2\,c\,d\,x\right )}^6}{256\,a^4\,c^4\,d^3-256\,a^3\,b^2\,c^3\,d^3+96\,a^2\,b^4\,c^2\,d^3-16\,a\,b^6\,c\,d^3+b^8\,d^3}-\frac {2106\,c^2\,{\left (b\,d+2\,c\,d\,x\right )}^4}{5\,\left (-64\,d\,a^3\,c^3+48\,d\,a^2\,b^2\,c^2-12\,d\,a\,b^4\,c+d\,b^6\right )}-\frac {64\,c^2\,d^3}{5\,\left (4\,a\,c-b^2\right )}+\frac {832\,c^2\,d\,{\left (b\,d+2\,c\,d\,x\right )}^2}{5\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{{\left (b\,d+2\,c\,d\,x\right )}^{5/2}\,\left (16\,a^2\,c^2\,d^4-8\,a\,b^2\,c\,d^4+b^4\,d^4\right )-{\left (b\,d+2\,c\,d\,x\right )}^{9/2}\,\left (2\,b^2\,d^2-8\,a\,c\,d^2\right )+{\left (b\,d+2\,c\,d\,x\right )}^{13/2}}+\frac {117\,c^2\,\mathrm {atan}\left (\frac {b^8\,\sqrt {b\,d+2\,c\,d\,x}+256\,a^4\,c^4\,\sqrt {b\,d+2\,c\,d\,x}+96\,a^2\,b^4\,c^2\,\sqrt {b\,d+2\,c\,d\,x}-256\,a^3\,b^2\,c^3\,\sqrt {b\,d+2\,c\,d\,x}-16\,a\,b^6\,c\,\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{17/4}}\right )}{d^{7/2}\,{\left (b^2-4\,a\,c\right )}^{17/4}}+\frac {c^2\,\mathrm {atan}\left (\frac {b^8\,\sqrt {b\,d+2\,c\,d\,x}\,1{}\mathrm {i}+a^4\,c^4\,\sqrt {b\,d+2\,c\,d\,x}\,256{}\mathrm {i}+a^2\,b^4\,c^2\,\sqrt {b\,d+2\,c\,d\,x}\,96{}\mathrm {i}-a^3\,b^2\,c^3\,\sqrt {b\,d+2\,c\,d\,x}\,256{}\mathrm {i}-a\,b^6\,c\,\sqrt {b\,d+2\,c\,d\,x}\,16{}\mathrm {i}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{17/4}}\right )\,117{}\mathrm {i}}{d^{7/2}\,{\left (b^2-4\,a\,c\right )}^{17/4}} \]

[In]

int(1/((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^3),x)

[Out]

((234*c^2*(b*d + 2*c*d*x)^6)/(b^8*d^3 + 256*a^4*c^4*d^3 + 96*a^2*b^4*c^2*d^3 - 256*a^3*b^2*c^3*d^3 - 16*a*b^6*
c*d^3) - (2106*c^2*(b*d + 2*c*d*x)^4)/(5*(b^6*d - 64*a^3*c^3*d + 48*a^2*b^2*c^2*d - 12*a*b^4*c*d)) - (64*c^2*d
^3)/(5*(4*a*c - b^2)) + (832*c^2*d*(b*d + 2*c*d*x)^2)/(5*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/((b*d + 2*c*d*x)^(5/
2)*(b^4*d^4 + 16*a^2*c^2*d^4 - 8*a*b^2*c*d^4) - (b*d + 2*c*d*x)^(9/2)*(2*b^2*d^2 - 8*a*c*d^2) + (b*d + 2*c*d*x
)^(13/2)) + (117*c^2*atan((b^8*(b*d + 2*c*d*x)^(1/2) + 256*a^4*c^4*(b*d + 2*c*d*x)^(1/2) + 96*a^2*b^4*c^2*(b*d
+ 2*c*d*x)^(1/2) - 256*a^3*b^2*c^3*(b*d + 2*c*d*x)^(1/2) - 16*a*b^6*c*(b*d + 2*c*d*x)^(1/2))/(d^(1/2)*(b^2 -
4*a*c)^(17/4))))/(d^(7/2)*(b^2 - 4*a*c)^(17/4)) + (c^2*atan((b^8*(b*d + 2*c*d*x)^(1/2)*1i + a^4*c^4*(b*d + 2*c
*d*x)^(1/2)*256i + a^2*b^4*c^2*(b*d + 2*c*d*x)^(1/2)*96i - a^3*b^2*c^3*(b*d + 2*c*d*x)^(1/2)*256i - a*b^6*c*(b
*d + 2*c*d*x)^(1/2)*16i)/(d^(1/2)*(b^2 - 4*a*c)^(17/4)))*117i)/(d^(7/2)*(b^2 - 4*a*c)^(17/4))