\(\int \frac {1}{(b d+2 c d x)^{5/2} (a+b x+c x^2)^2} \, dx\) [1305]

Optimal result

Integrand size = 26, antiderivative size = 174 \[ \int \frac {1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2} \, dx=-\frac {28 c}{3 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2}}-\frac {1}{\left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )}+\frac {14 c \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{11/4} d^{5/2}}+\frac {14 c \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{11/4} d^{5/2}} \]

[Out]

-28/3*c/(-4*a*c+b^2)^2/d/(2*c*d*x+b*d)^(3/2)-1/(-4*a*c+b^2)/d/(2*c*d*x+b*d)^(3/2)/(c*x^2+b*x+a)+14*c*arctan((d
*(2*c*x+b))^(1/2)/(-4*a*c+b^2)^(1/4)/d^(1/2))/(-4*a*c+b^2)^(11/4)/d^(5/2)+14*c*arctanh((d*(2*c*x+b))^(1/2)/(-4
*a*c+b^2)^(1/4)/d^(1/2))/(-4*a*c+b^2)^(11/4)/d^(5/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {701, 707, 708, 335, 218, 212, 209} \[ \int \frac {1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2} \, dx=\frac {14 c \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d^{5/2} \left (b^2-4 a c\right )^{11/4}}+\frac {14 c \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d^{5/2} \left (b^2-4 a c\right )^{11/4}}-\frac {1}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) (b d+2 c d x)^{3/2}}-\frac {28 c}{3 d \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}} \]

[In]

Int[1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^2),x]

[Out]

(-28*c)/(3*(b^2 - 4*a*c)^2*d*(b*d + 2*c*d*x)^(3/2)) - 1/((b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^
2)) + (14*c*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^(11/4)*d^(5/2)) + (14*c*
ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^(11/4)*d^(5/2))

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 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[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}{\left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )}-\frac {(7 c) \int \frac {1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )} \, dx}{b^2-4 a c} \\ & = -\frac {28 c}{3 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2}}-\frac {1}{\left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )}-\frac {(7 c) \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right )^2 d^2} \\ & = -\frac {28 c}{3 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2}}-\frac {1}{\left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )}-\frac {7 \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}\right )} \, dx,x,b d+2 c d x\right )}{2 \left (b^2-4 a c\right )^2 d^3} \\ & = -\frac {28 c}{3 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2}}-\frac {1}{\left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )}-\frac {7 \text {Subst}\left (\int \frac {1}{a-\frac {b^2}{4 c}+\frac {x^4}{4 c d^2}} \, dx,x,\sqrt {d (b+2 c x)}\right )}{\left (b^2-4 a c\right )^2 d^3} \\ & = -\frac {28 c}{3 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2}}-\frac {1}{\left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )}+\frac {(14 c) \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 )^{5/2} d^2}+\frac {(14 c) \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 )^{5/2} d^2} \\ & = -\frac {28 c}{3 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2}}-\frac {1}{\left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )}+\frac {14 c \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 )^{11/4} d^{5/2}}+\frac {14 c \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 )^{11/4} d^{5/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.69 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.54 \[ \int \frac {1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2} \, dx=\frac {\left (\frac {1}{3}+\frac {i}{3}\right ) c \left (-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) (b+2 c x) \left (-4 b^2+16 a c+7 (b+2 c x)^2\right )}{c \left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac {21 i (b+2 c x)^{5/2} \arctan \left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{11/4}}-\frac {21 i (b+2 c x)^{5/2} \arctan \left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{11/4}}-\frac {21 i (b+2 c x)^{5/2} \text {arctanh}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )}{\left (b^2-4 a c\right )^{11/4}}\right )}{(d (b+2 c x))^{5/2}} \]

[In]

Integrate[1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^2),x]

[Out]

((1/3 + I/3)*c*(((-1/2 + I/2)*(b + 2*c*x)*(-4*b^2 + 16*a*c + 7*(b + 2*c*x)^2))/(c*(b^2 - 4*a*c)^2*(a + x*(b +
c*x))) + ((21*I)*(b + 2*c*x)^(5/2)*ArcTan[1 - ((1 + I)*Sqrt[b + 2*c*x])/(b^2 - 4*a*c)^(1/4)])/(b^2 - 4*a*c)^(1
1/4) - ((21*I)*(b + 2*c*x)^(5/2)*ArcTan[1 + ((1 + I)*Sqrt[b + 2*c*x])/(b^2 - 4*a*c)^(1/4)])/(b^2 - 4*a*c)^(11/
4) - ((21*I)*(b + 2*c*x)^(5/2)*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)^(11/4)))/(d*(b + 2*c*x))^(5/2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(322\) vs. \(2(148)=296\).

Time = 2.97 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.86

method	result	size
derivativedivides	\(16 c \,d^{3} \left (-\frac {\frac {\sqrt {2 c d x +b d}}{16 a c \,d^{2}-4 b^{2} d^{2}+4 \left (2 c d x +b d \right )^{2}}+\frac {7 \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{32 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}}{d^{4} \left (4 a c -b^{2}\right )^{2}}-\frac {1}{3 d^{4} \left (4 a c -b^{2}\right )^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}}\right )\)	\(323\)
default	\(16 c \,d^{3} \left (-\frac {\frac {\sqrt {2 c d x +b d}}{16 a c \,d^{2}-4 b^{2} d^{2}+4 \left (2 c d x +b d \right )^{2}}+\frac {7 \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{32 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}}{d^{4} \left (4 a c -b^{2}\right )^{2}}-\frac {1}{3 d^{4} \left (4 a c -b^{2}\right )^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}}\right )\)	\(323\)
pseudoelliptic	\(\frac {8 d^{3} c \left (-\frac {\sqrt {d \left (2 c x +b \right )}}{8 c \,d^{6} \left (c \,x^{2}+b x +a \right )}-\frac {7 \ln \left (\frac {\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+\sqrt {d^{2} \left (4 a c -b^{2}\right )}+d \left (2 c x +b \right )}{\sqrt {d^{2} \left (4 a c -b^{2}\right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+d \left (2 c x +b \right )}\right ) \sqrt {2}}{16 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} d^{4}}-\frac {7 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right ) \sqrt {2}}{8 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} d^{4}}+\frac {7 \arctan \left (\frac {-\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right ) \sqrt {2}}{8 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} d^{4}}-\frac {2}{3 d^{4} \left (d \left (2 c x +b \right )\right )^{\frac {3}{2}}}\right )}{\left (4 a c -b^{2}\right )^{2}}\)	\(351\)

[In]

int(1/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

16*c*d^3*(-1/d^4/(4*a*c-b^2)^2*(1/16*(2*c*d*x+b*d)^(1/2)/(a*c*d^2-1/4*b^2*d^2+1/4*(2*c*d*x+b*d)^2)+7/32/(4*a*c
*d^2-b^2*d^2)^(3/4)*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)))-1/3/d^4/(4*a*c-b^2)^2/(2*c*d*x+b*d)^(3/2))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 2171, normalized size of antiderivative = 12.48 \[ \int \frac {1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate(1/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

1/3*(21*(4*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3*x^4 + 8*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^3*x^3 + (
5*b^6*c - 36*a*b^4*c^2 + 48*a^2*b^2*c^3 + 64*a^3*c^4)*d^3*x^2 + (b^7 - 4*a*b^5*c - 16*a^2*b^3*c^2 + 64*a^3*b*c
^3)*d^3*x + (a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*d^3)*(c^4/((b^22 - 44*a*b^20*c + 880*a^2*b^18*c^2 - 10560*a
^3*b^16*c^3 + 84480*a^4*b^14*c^4 - 473088*a^5*b^12*c^5 + 1892352*a^6*b^10*c^6 - 5406720*a^7*b^8*c^7 + 10813440
*a^8*b^6*c^8 - 14417920*a^9*b^4*c^9 + 11534336*a^10*b^2*c^10 - 4194304*a^11*c^11)*d^10))^(1/4)*log(7*(b^6 - 12
*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*d^3*(c^4/((b^22 - 44*a*b^20*c + 880*a^2*b^18*c^2 - 10560*a^3*b^16*c^3
+ 84480*a^4*b^14*c^4 - 473088*a^5*b^12*c^5 + 1892352*a^6*b^10*c^6 - 5406720*a^7*b^8*c^7 + 10813440*a^8*b^6*c^8
- 14417920*a^9*b^4*c^9 + 11534336*a^10*b^2*c^10 - 4194304*a^11*c^11)*d^10))^(1/4) + 7*sqrt(2*c*d*x + b*d)*c)
- 21*(-4*I*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3*x^4 - 8*I*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^3*x^3 -
I*(5*b^6*c - 36*a*b^4*c^2 + 48*a^2*b^2*c^3 + 64*a^3*c^4)*d^3*x^2 - I*(b^7 - 4*a*b^5*c - 16*a^2*b^3*c^2 + 64*a
^3*b*c^3)*d^3*x - I*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*d^3)*(c^4/((b^22 - 44*a*b^20*c + 880*a^2*b^18*c^2 -
10560*a^3*b^16*c^3 + 84480*a^4*b^14*c^4 - 473088*a^5*b^12*c^5 + 1892352*a^6*b^10*c^6 - 5406720*a^7*b^8*c^7 +
10813440*a^8*b^6*c^8 - 14417920*a^9*b^4*c^9 + 11534336*a^10*b^2*c^10 - 4194304*a^11*c^11)*d^10))^(1/4)*log(7*I
*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*d^3*(c^4/((b^22 - 44*a*b^20*c + 880*a^2*b^18*c^2 - 10560*a^3
*b^16*c^3 + 84480*a^4*b^14*c^4 - 473088*a^5*b^12*c^5 + 1892352*a^6*b^10*c^6 - 5406720*a^7*b^8*c^7 + 10813440*a
^8*b^6*c^8 - 14417920*a^9*b^4*c^9 + 11534336*a^10*b^2*c^10 - 4194304*a^11*c^11)*d^10))^(1/4) + 7*sqrt(2*c*d*x
+ b*d)*c) - 21*(4*I*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3*x^4 + 8*I*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*
d^3*x^3 + I*(5*b^6*c - 36*a*b^4*c^2 + 48*a^2*b^2*c^3 + 64*a^3*c^4)*d^3*x^2 + I*(b^7 - 4*a*b^5*c - 16*a^2*b^3*c
^2 + 64*a^3*b*c^3)*d^3*x + I*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*d^3)*(c^4/((b^22 - 44*a*b^20*c + 880*a^2*b
^18*c^2 - 10560*a^3*b^16*c^3 + 84480*a^4*b^14*c^4 - 473088*a^5*b^12*c^5 + 1892352*a^6*b^10*c^6 - 5406720*a^7*b
^8*c^7 + 10813440*a^8*b^6*c^8 - 14417920*a^9*b^4*c^9 + 11534336*a^10*b^2*c^10 - 4194304*a^11*c^11)*d^10))^(1/4
)*log(-7*I*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*d^3*(c^4/((b^22 - 44*a*b^20*c + 880*a^2*b^18*c^2 -
10560*a^3*b^16*c^3 + 84480*a^4*b^14*c^4 - 473088*a^5*b^12*c^5 + 1892352*a^6*b^10*c^6 - 5406720*a^7*b^8*c^7 +
10813440*a^8*b^6*c^8 - 14417920*a^9*b^4*c^9 + 11534336*a^10*b^2*c^10 - 4194304*a^11*c^11)*d^10))^(1/4) + 7*sqr
t(2*c*d*x + b*d)*c) - 21*(4*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3*x^4 + 8*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b
*c^4)*d^3*x^3 + (5*b^6*c - 36*a*b^4*c^2 + 48*a^2*b^2*c^3 + 64*a^3*c^4)*d^3*x^2 + (b^7 - 4*a*b^5*c - 16*a^2*b^3
*c^2 + 64*a^3*b*c^3)*d^3*x + (a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*d^3)*(c^4/((b^22 - 44*a*b^20*c + 880*a^2*b
^18*c^2 - 10560*a^3*b^16*c^3 + 84480*a^4*b^14*c^4 - 473088*a^5*b^12*c^5 + 1892352*a^6*b^10*c^6 - 5406720*a^7*b
^8*c^7 + 10813440*a^8*b^6*c^8 - 14417920*a^9*b^4*c^9 + 11534336*a^10*b^2*c^10 - 4194304*a^11*c^11)*d^10))^(1/4
)*log(-7*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*d^3*(c^4/((b^22 - 44*a*b^20*c + 880*a^2*b^18*c^2 - 1
0560*a^3*b^16*c^3 + 84480*a^4*b^14*c^4 - 473088*a^5*b^12*c^5 + 1892352*a^6*b^10*c^6 - 5406720*a^7*b^8*c^7 + 10
813440*a^8*b^6*c^8 - 14417920*a^9*b^4*c^9 + 11534336*a^10*b^2*c^10 - 4194304*a^11*c^11)*d^10))^(1/4) + 7*sqrt(
2*c*d*x + b*d)*c) - (28*c^2*x^2 + 28*b*c*x + 3*b^2 + 16*a*c)*sqrt(2*c*d*x + b*d))/(4*(b^4*c^3 - 8*a*b^2*c^4 +
16*a^2*c^5)*d^3*x^4 + 8*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^3*x^3 + (5*b^6*c - 36*a*b^4*c^2 + 48*a^2*b^2*
c^3 + 64*a^3*c^4)*d^3*x^2 + (b^7 - 4*a*b^5*c - 16*a^2*b^3*c^2 + 64*a^3*b*c^3)*d^3*x + (a*b^6 - 8*a^2*b^4*c + 1
6*a^3*b^2*c^2)*d^3)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]

[In]

integrate(1/(2*c*d*x+b*d)**(5/2)/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^2,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. 646 vs. \(2 (148) = 296\).

Time = 0.30 (sec) , antiderivative size = 646, normalized size of antiderivative = 3.71 \[ \int \frac {1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2} \, dx=\frac {7 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c \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 )}{b^{6} d^{3} - 12 \, a b^{4} c d^{3} + 48 \, a^{2} b^{2} c^{2} d^{3} - 64 \, a^{3} c^{3} d^{3}} + \frac {7 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c \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 )}{b^{6} d^{3} - 12 \, a b^{4} c d^{3} + 48 \, a^{2} b^{2} c^{2} d^{3} - 64 \, a^{3} c^{3} d^{3}} + \frac {7 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c \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 )}{\sqrt {2} b^{6} d^{3} - 12 \, \sqrt {2} a b^{4} c d^{3} + 48 \, \sqrt {2} a^{2} b^{2} c^{2} d^{3} - 64 \, \sqrt {2} a^{3} c^{3} d^{3}} - \frac {7 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c \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 )}{\sqrt {2} b^{6} d^{3} - 12 \, \sqrt {2} a b^{4} c d^{3} + 48 \, \sqrt {2} a^{2} b^{2} c^{2} d^{3} - 64 \, \sqrt {2} a^{3} c^{3} d^{3}} + \frac {4 \, \sqrt {2 \, c d x + b d} c}{{\left (b^{4} d - 8 \, a b^{2} c d + 16 \, a^{2} c^{2} d\right )} {\left (b^{2} d^{2} - 4 \, a c d^{2} - {\left (2 \, c d x + b d\right )}^{2}\right )}} - \frac {16 \, c}{3 \, {\left (b^{4} d - 8 \, a b^{2} c d + 16 \, a^{2} c^{2} d\right )} {\left (2 \, c d x + b d\right )}^{\frac {3}{2}}} \]

[In]

integrate(1/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

7*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*c*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))/(b^6*d^3 - 12*a*b^4*c*d^3 + 48*a^2*b^2*c^2*d^3 - 64*a^3*c^3*d^3) +
7*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*c*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))/(b^6*d^3 - 12*a*b^4*c*d^3 + 48*a^2*b^2*c^2*d^3 - 64*a^3*c^3*d^3) +
7*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*c*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^6*d^3 - 12*sqrt(2)*a*b^4*c*d^3 + 48*sqrt(2)*a^2*b^2*c^2*d^3 - 64*sqr
t(2)*a^3*c^3*d^3) - 7*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*c*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^6*d^3 - 12*sqrt(2)*a*b^4*c*d^3 + 48*sqrt(2)*a^2*b
^2*c^2*d^3 - 64*sqrt(2)*a^3*c^3*d^3) + 4*sqrt(2*c*d*x + b*d)*c/((b^4*d - 8*a*b^2*c*d + 16*a^2*c^2*d)*(b^2*d^2
- 4*a*c*d^2 - (2*c*d*x + b*d)^2)) - 16/3*c/((b^4*d - 8*a*b^2*c*d + 16*a^2*c^2*d)*(2*c*d*x + b*d)^(3/2))

Mupad [B] (verification not implemented)

Time = 9.53 (sec) , antiderivative size = 2280, normalized size of antiderivative = 13.10 \[ \int \frac {1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

int(1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^2),x)

[Out]

((16*c*d)/(3*(4*a*c - b^2)) + (28*c*(b*d + 2*c*d*x)^2)/(3*d*(4*a*c - b^2)^2))/((b*d + 2*c*d*x)^(3/2)*(b^2*d^2
- 4*a*c*d^2) - (b*d + 2*c*d*x)^(7/2)) + (c*atan(((c*((b*d + 2*c*d*x)^(1/2)*(12845056*a^6*c^8*d^3 + 3136*b^12*c
^2*d^3 - 75264*a*b^10*c^3*d^3 + 752640*a^2*b^8*c^4*d^3 - 4014080*a^3*b^6*c^5*d^3 + 12042240*a^4*b^4*c^6*d^3 -
19267584*a^5*b^2*c^7*d^3) - (7*c*(448*b^18*c*d^6 - 117440512*a^9*c^10*d^6 - 16128*a*b^16*c^2*d^6 + 258048*a^2*
b^14*c^3*d^6 - 2408448*a^3*b^12*c^4*d^6 + 14450688*a^4*b^10*c^5*d^6 - 57802752*a^5*b^8*c^6*d^6 + 154140672*a^6
*b^6*c^7*d^6 - 264241152*a^7*b^4*c^8*d^6 + 264241152*a^8*b^2*c^9*d^6))/(d^(5/2)*(b^2 - 4*a*c)^(11/4)))*7i)/(d^
(5/2)*(b^2 - 4*a*c)^(11/4)) + (c*((b*d + 2*c*d*x)^(1/2)*(12845056*a^6*c^8*d^3 + 3136*b^12*c^2*d^3 - 75264*a*b^
10*c^3*d^3 + 752640*a^2*b^8*c^4*d^3 - 4014080*a^3*b^6*c^5*d^3 + 12042240*a^4*b^4*c^6*d^3 - 19267584*a^5*b^2*c^
7*d^3) + (7*c*(448*b^18*c*d^6 - 117440512*a^9*c^10*d^6 - 16128*a*b^16*c^2*d^6 + 258048*a^2*b^14*c^3*d^6 - 2408
448*a^3*b^12*c^4*d^6 + 14450688*a^4*b^10*c^5*d^6 - 57802752*a^5*b^8*c^6*d^6 + 154140672*a^6*b^6*c^7*d^6 - 2642
41152*a^7*b^4*c^8*d^6 + 264241152*a^8*b^2*c^9*d^6))/(d^(5/2)*(b^2 - 4*a*c)^(11/4)))*7i)/(d^(5/2)*(b^2 - 4*a*c)
^(11/4)))/((7*c*((b*d + 2*c*d*x)^(1/2)*(12845056*a^6*c^8*d^3 + 3136*b^12*c^2*d^3 - 75264*a*b^10*c^3*d^3 + 7526
40*a^2*b^8*c^4*d^3 - 4014080*a^3*b^6*c^5*d^3 + 12042240*a^4*b^4*c^6*d^3 - 19267584*a^5*b^2*c^7*d^3) - (7*c*(44
8*b^18*c*d^6 - 117440512*a^9*c^10*d^6 - 16128*a*b^16*c^2*d^6 + 258048*a^2*b^14*c^3*d^6 - 2408448*a^3*b^12*c^4*
d^6 + 14450688*a^4*b^10*c^5*d^6 - 57802752*a^5*b^8*c^6*d^6 + 154140672*a^6*b^6*c^7*d^6 - 264241152*a^7*b^4*c^8
*d^6 + 264241152*a^8*b^2*c^9*d^6))/(d^(5/2)*(b^2 - 4*a*c)^(11/4))))/(d^(5/2)*(b^2 - 4*a*c)^(11/4)) - (7*c*((b*
d + 2*c*d*x)^(1/2)*(12845056*a^6*c^8*d^3 + 3136*b^12*c^2*d^3 - 75264*a*b^10*c^3*d^3 + 752640*a^2*b^8*c^4*d^3 -
4014080*a^3*b^6*c^5*d^3 + 12042240*a^4*b^4*c^6*d^3 - 19267584*a^5*b^2*c^7*d^3) + (7*c*(448*b^18*c*d^6 - 11744
0512*a^9*c^10*d^6 - 16128*a*b^16*c^2*d^6 + 258048*a^2*b^14*c^3*d^6 - 2408448*a^3*b^12*c^4*d^6 + 14450688*a^4*b
^10*c^5*d^6 - 57802752*a^5*b^8*c^6*d^6 + 154140672*a^6*b^6*c^7*d^6 - 264241152*a^7*b^4*c^8*d^6 + 264241152*a^8
*b^2*c^9*d^6))/(d^(5/2)*(b^2 - 4*a*c)^(11/4))))/(d^(5/2)*(b^2 - 4*a*c)^(11/4))))*14i)/(d^(5/2)*(b^2 - 4*a*c)^(
11/4)) + (14*c*atan(((7*c*((b*d + 2*c*d*x)^(1/2)*(12845056*a^6*c^8*d^3 + 3136*b^12*c^2*d^3 - 75264*a*b^10*c^3*
d^3 + 752640*a^2*b^8*c^4*d^3 - 4014080*a^3*b^6*c^5*d^3 + 12042240*a^4*b^4*c^6*d^3 - 19267584*a^5*b^2*c^7*d^3)
- (c*(448*b^18*c*d^6 - 117440512*a^9*c^10*d^6 - 16128*a*b^16*c^2*d^6 + 258048*a^2*b^14*c^3*d^6 - 2408448*a^3*b
^12*c^4*d^6 + 14450688*a^4*b^10*c^5*d^6 - 57802752*a^5*b^8*c^6*d^6 + 154140672*a^6*b^6*c^7*d^6 - 264241152*a^7
*b^4*c^8*d^6 + 264241152*a^8*b^2*c^9*d^6)*7i)/(d^(5/2)*(b^2 - 4*a*c)^(11/4))))/(d^(5/2)*(b^2 - 4*a*c)^(11/4))
+ (7*c*((b*d + 2*c*d*x)^(1/2)*(12845056*a^6*c^8*d^3 + 3136*b^12*c^2*d^3 - 75264*a*b^10*c^3*d^3 + 752640*a^2*b^
8*c^4*d^3 - 4014080*a^3*b^6*c^5*d^3 + 12042240*a^4*b^4*c^6*d^3 - 19267584*a^5*b^2*c^7*d^3) + (c*(448*b^18*c*d^
6 - 117440512*a^9*c^10*d^6 - 16128*a*b^16*c^2*d^6 + 258048*a^2*b^14*c^3*d^6 - 2408448*a^3*b^12*c^4*d^6 + 14450
688*a^4*b^10*c^5*d^6 - 57802752*a^5*b^8*c^6*d^6 + 154140672*a^6*b^6*c^7*d^6 - 264241152*a^7*b^4*c^8*d^6 + 2642
41152*a^8*b^2*c^9*d^6)*7i)/(d^(5/2)*(b^2 - 4*a*c)^(11/4))))/(d^(5/2)*(b^2 - 4*a*c)^(11/4)))/((c*((b*d + 2*c*d*
x)^(1/2)*(12845056*a^6*c^8*d^3 + 3136*b^12*c^2*d^3 - 75264*a*b^10*c^3*d^3 + 752640*a^2*b^8*c^4*d^3 - 4014080*a
^3*b^6*c^5*d^3 + 12042240*a^4*b^4*c^6*d^3 - 19267584*a^5*b^2*c^7*d^3) - (c*(448*b^18*c*d^6 - 117440512*a^9*c^1
0*d^6 - 16128*a*b^16*c^2*d^6 + 258048*a^2*b^14*c^3*d^6 - 2408448*a^3*b^12*c^4*d^6 + 14450688*a^4*b^10*c^5*d^6
- 57802752*a^5*b^8*c^6*d^6 + 154140672*a^6*b^6*c^7*d^6 - 264241152*a^7*b^4*c^8*d^6 + 264241152*a^8*b^2*c^9*d^6
)*7i)/(d^(5/2)*(b^2 - 4*a*c)^(11/4)))*7i)/(d^(5/2)*(b^2 - 4*a*c)^(11/4)) - (c*((b*d + 2*c*d*x)^(1/2)*(12845056
*a^6*c^8*d^3 + 3136*b^12*c^2*d^3 - 75264*a*b^10*c^3*d^3 + 752640*a^2*b^8*c^4*d^3 - 4014080*a^3*b^6*c^5*d^3 + 1
2042240*a^4*b^4*c^6*d^3 - 19267584*a^5*b^2*c^7*d^3) + (c*(448*b^18*c*d^6 - 117440512*a^9*c^10*d^6 - 16128*a*b^
16*c^2*d^6 + 258048*a^2*b^14*c^3*d^6 - 2408448*a^3*b^12*c^4*d^6 + 14450688*a^4*b^10*c^5*d^6 - 57802752*a^5*b^8
*c^6*d^6 + 154140672*a^6*b^6*c^7*d^6 - 264241152*a^7*b^4*c^8*d^6 + 264241152*a^8*b^2*c^9*d^6)*7i)/(d^(5/2)*(b^
2 - 4*a*c)^(11/4)))*7i)/(d^(5/2)*(b^2 - 4*a*c)^(11/4)))))/(d^(5/2)*(b^2 - 4*a*c)^(11/4))