\(\int \frac {(b d+2 c d x)^{15/2}}{(a+b x+c x^2)^3} \, dx\) [1308]

Optimal result

Integrand size = 26, antiderivative size = 222 \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^3} \, dx=234 c^2 \left (b^2-4 a c\right ) d^7 \sqrt {b d+2 c d x}+\frac {234}{5} c^2 d^5 (b d+2 c d x)^{5/2}-\frac {d (b d+2 c d x)^{13/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {13 c d^3 (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )}-117 c^2 \left (b^2-4 a c\right )^{5/4} d^{15/2} \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-117 c^2 \left (b^2-4 a c\right )^{5/4} d^{15/2} \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ) \]

[Out]

234/5*c^2*d^5*(2*c*d*x+b*d)^(5/2)-1/2*d*(2*c*d*x+b*d)^(13/2)/(c*x^2+b*x+a)^2-13/2*c*d^3*(2*c*d*x+b*d)^(9/2)/(c
*x^2+b*x+a)-117*c^2*(-4*a*c+b^2)^(5/4)*d^(15/2)*arctan((d*(2*c*x+b))^(1/2)/(-4*a*c+b^2)^(1/4)/d^(1/2))-117*c^2
*(-4*a*c+b^2)^(5/4)*d^(15/2)*arctanh((d*(2*c*x+b))^(1/2)/(-4*a*c+b^2)^(1/4)/d^(1/2))+234*c^2*(-4*a*c+b^2)*d^7*
(2*c*d*x+b*d)^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 222, 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 = {700, 706, 708, 335, 218, 212, 209} \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^3} \, dx=-117 c^2 d^{15/2} \left (b^2-4 a c\right )^{5/4} \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-117 c^2 d^{15/2} \left (b^2-4 a c\right )^{5/4} \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )+234 c^2 d^7 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}-\frac {13 c d^3 (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )}-\frac {d (b d+2 c d x)^{13/2}}{2 \left (a+b x+c x^2\right )^2}+\frac {234}{5} c^2 d^5 (b d+2 c d x)^{5/2} \]

[In]

Int[(b*d + 2*c*d*x)^(15/2)/(a + b*x + c*x^2)^3,x]

[Out]

234*c^2*(b^2 - 4*a*c)*d^7*Sqrt[b*d + 2*c*d*x] + (234*c^2*d^5*(b*d + 2*c*d*x)^(5/2))/5 - (d*(b*d + 2*c*d*x)^(13
/2))/(2*(a + b*x + c*x^2)^2) - (13*c*d^3*(b*d + 2*c*d*x)^(9/2))/(2*(a + b*x + c*x^2)) - 117*c^2*(b^2 - 4*a*c)^
(5/4)*d^(15/2)*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])] - 117*c^2*(b^2 - 4*a*c)^(5/4)*d^(15/2
)*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])]

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 700

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

Rule 706

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2*d*(d + e*x)^(m - 1
)*((a + b*x + c*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] + Dist[d^2*(m - 1)*((b^2 - 4*a*c)/(b^2*(m + 2*p + 1))), In
t[(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] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] &
& RationalQ[p]) || OddQ[m])

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 {d (b d+2 c d x)^{13/2}}{2 \left (a+b x+c x^2\right )^2}+\frac {1}{2} \left (13 c d^2\right ) \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^2} \, dx \\ & = -\frac {d (b d+2 c d x)^{13/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {13 c d^3 (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )}+\frac {1}{2} \left (117 c^2 d^4\right ) \int \frac {(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx \\ & = \frac {234}{5} c^2 d^5 (b d+2 c d x)^{5/2}-\frac {d (b d+2 c d x)^{13/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {13 c d^3 (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )}+\frac {1}{2} \left (117 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac {(b d+2 c d x)^{3/2}}{a+b x+c x^2} \, dx \\ & = 234 c^2 \left (b^2-4 a c\right ) d^7 \sqrt {b d+2 c d x}+\frac {234}{5} c^2 d^5 (b d+2 c d x)^{5/2}-\frac {d (b d+2 c d x)^{13/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {13 c d^3 (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )}+\frac {1}{2} \left (117 c^2 \left (b^2-4 a c\right )^2 d^8\right ) \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )} \, dx \\ & = 234 c^2 \left (b^2-4 a c\right ) d^7 \sqrt {b d+2 c d x}+\frac {234}{5} c^2 d^5 (b d+2 c d x)^{5/2}-\frac {d (b d+2 c d x)^{13/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {13 c d^3 (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )}+\frac {1}{4} \left (117 c \left (b^2-4 a c\right )^2 d^7\right ) \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 ) \\ & = 234 c^2 \left (b^2-4 a c\right ) d^7 \sqrt {b d+2 c d x}+\frac {234}{5} c^2 d^5 (b d+2 c d x)^{5/2}-\frac {d (b d+2 c d x)^{13/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {13 c d^3 (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )}+\frac {1}{2} \left (117 c \left (b^2-4 a c\right )^2 d^7\right ) \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 ) \\ & = 234 c^2 \left (b^2-4 a c\right ) d^7 \sqrt {b d+2 c d x}+\frac {234}{5} c^2 d^5 (b d+2 c d x)^{5/2}-\frac {d (b d+2 c d x)^{13/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {13 c d^3 (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )}-\left (117 c^2 \left (b^2-4 a c\right )^{3/2} d^8\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 (117 c^2 \left (b^2-4 a c\right )^{3/2} d^8\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 ) \\ & = 234 c^2 \left (b^2-4 a c\right ) d^7 \sqrt {b d+2 c d x}+\frac {234}{5} c^2 d^5 (b d+2 c d x)^{5/2}-\frac {d (b d+2 c d x)^{13/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {13 c d^3 (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )}-117 c^2 \left (b^2-4 a c\right )^{5/4} d^{15/2} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-117 c^2 \left (b^2-4 a c\right )^{5/4} d^{15/2} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.86 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.59 \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^3} \, dx=\left (\frac {1}{10}+\frac {i}{10}\right ) c^2 (d (b+2 c x))^{15/2} \left (-\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \left (-585 b^6+7020 a b^4 c-28080 a^2 b^2 c^2+37440 a^3 c^3+1053 b^4 (b+2 c x)^2-8424 a b^2 c (b+2 c x)^2+16848 a^2 c^2 (b+2 c x)^2-416 b^2 (b+2 c x)^4+1664 a c (b+2 c x)^4-32 (b+2 c x)^6\right )}{c^2 (b+2 c x)^7 (a+x (b+c x))^2}-\frac {585 i \left (b^2-4 a c\right )^{5/4} \arctan \left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{15/2}}+\frac {585 i \left (b^2-4 a c\right )^{5/4} \arctan \left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{15/2}}+\frac {585 i \left (b^2-4 a c\right )^{5/4} \text {arctanh}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )}{(b+2 c x)^{15/2}}\right ) \]

[In]

Integrate[(b*d + 2*c*d*x)^(15/2)/(a + b*x + c*x^2)^3,x]

[Out]

(1/10 + I/10)*c^2*(d*(b + 2*c*x))^(15/2)*(((-1/8 + I/8)*(-585*b^6 + 7020*a*b^4*c - 28080*a^2*b^2*c^2 + 37440*a
^3*c^3 + 1053*b^4*(b + 2*c*x)^2 - 8424*a*b^2*c*(b + 2*c*x)^2 + 16848*a^2*c^2*(b + 2*c*x)^2 - 416*b^2*(b + 2*c*
x)^4 + 1664*a*c*(b + 2*c*x)^4 - 32*(b + 2*c*x)^6))/(c^2*(b + 2*c*x)^7*(a + x*(b + c*x))^2) - ((585*I)*(b^2 - 4
*a*c)^(5/4)*ArcTan[1 - ((1 + I)*Sqrt[b + 2*c*x])/(b^2 - 4*a*c)^(1/4)])/(b + 2*c*x)^(15/2) + ((585*I)*(b^2 - 4*
a*c)^(5/4)*ArcTan[1 + ((1 + I)*Sqrt[b + 2*c*x])/(b^2 - 4*a*c)^(1/4)])/(b + 2*c*x)^(15/2) + ((585*I)*(b^2 - 4*a
*c)^(5/4)*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*c
*x)^(15/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(431\) vs. \(2(188)=376\).

Time = 2.80 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.95

method	result	size
derivativedivides	\(64 c^{2} d^{5} \left (-12 a c \,d^{2} \sqrt {2 c d x +b d}+3 b^{2} d^{2} \sqrt {2 c d x +b d}+\frac {\left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+d^{4} \left (\frac {16 \left (-\frac {25}{32} a^{2} c^{2}+\frac {25}{64} a \,b^{2} c -\frac {25}{512} b^{4}\right ) \left (2 c d x +b d \right )^{\frac {5}{2}}+16 \left (-\frac {21}{8} a^{3} c^{3} d^{2}+\frac {63}{32} a^{2} b^{2} c^{2} d^{2}-\frac {63}{128} a \,b^{4} c \,d^{2}+\frac {21}{512} b^{6} d^{2}\right ) \sqrt {2 c d x +b d}}{\left (\left (2 c d x +b d \right )^{2}+4 a c \,d^{2}-b^{2} d^{2}\right )^{2}}+\frac {117 \left (a^{2} c^{2}-\frac {1}{2} a \,b^{2} c +\frac {1}{16} b^{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\right )\)	\(432\)
default	\(64 c^{2} d^{5} \left (-12 a c \,d^{2} \sqrt {2 c d x +b d}+3 b^{2} d^{2} \sqrt {2 c d x +b d}+\frac {\left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+d^{4} \left (\frac {16 \left (-\frac {25}{32} a^{2} c^{2}+\frac {25}{64} a \,b^{2} c -\frac {25}{512} b^{4}\right ) \left (2 c d x +b d \right )^{\frac {5}{2}}+16 \left (-\frac {21}{8} a^{3} c^{3} d^{2}+\frac {63}{32} a^{2} b^{2} c^{2} d^{2}-\frac {63}{128} a \,b^{4} c \,d^{2}+\frac {21}{512} b^{6} d^{2}\right ) \sqrt {2 c d x +b d}}{\left (\left (2 c d x +b d \right )^{2}+4 a c \,d^{2}-b^{2} d^{2}\right )^{2}}+\frac {117 \left (a^{2} c^{2}-\frac {1}{2} a \,b^{2} c +\frac {1}{16} b^{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\right )\)	\(432\)
pseudoelliptic	\(-\frac {234 \left (\frac {31 \left (-\frac {32 c^{4} x^{4}}{93}-\frac {64 x^{2} \left (b x +a \right ) c^{3}}{93}+\left (a^{2}-\frac {64}{93} a b x -\frac {32}{93} b^{2} x^{2}\right ) c^{2}-\frac {125 a \,b^{2} c}{186}+\frac {125 b^{4}}{1488}\right ) \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} \left (d \left (2 c x +b \right )\right )^{\frac {5}{2}}}{195}+\frac {d^{2} \left (4 a c -b^{2}\right ) \left (8 \left (\frac {32 c^{4} x^{4}}{39}+\frac {64 x^{2} \left (b x +a \right ) c^{3}}{39}+\left (\frac {32}{39} b^{2} x^{2}+\frac {64}{39} a b x +a^{2}\right ) c^{2}-\frac {7 a \,b^{2} c}{78}+\frac {7 b^{4}}{624}\right ) \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} \sqrt {d \left (2 c x +b \right )}+c^{2} d^{2} \sqrt {2}\, \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{2} \left (2 \arctan \left (\frac {-\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right )-2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+\sqrt {d^{2} \left (4 a c -b^{2}\right )}+d \left (2 c x +b \right )}{\sqrt {d^{2} \left (4 a c -b^{2}\right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+d \left (2 c x +b \right )}\right )\right )\right )}{8}\right ) d^{5}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} \left (c \,x^{2}+b x +a \right )^{2}}\)	\(470\)

[In]

int((2*c*d*x+b*d)^(15/2)/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

64*c^2*d^5*(-12*a*c*d^2*(2*c*d*x+b*d)^(1/2)+3*b^2*d^2*(2*c*d*x+b*d)^(1/2)+1/5*(2*c*d*x+b*d)^(5/2)+d^4*(16*((-2
5/32*a^2*c^2+25/64*a*b^2*c-25/512*b^4)*(2*c*d*x+b*d)^(5/2)+(-21/8*a^3*c^3*d^2+63/32*a^2*b^2*c^2*d^2-63/128*a*b
^4*c*d^2+21/512*b^6*d^2)*(2*c*d*x+b*d)^(1/2))/((2*c*d*x+b*d)^2+4*a*c*d^2-b^2*d^2)^2+117/16*(a^2*c^2-1/2*a*b^2*
c+1/16*b^4)/(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))))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 1029, normalized size of antiderivative = 4.64 \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate((2*c*d*x+b*d)^(15/2)/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

1/10*(585*((b^10*c^8 - 20*a*b^8*c^9 + 160*a^2*b^6*c^10 - 640*a^3*b^4*c^11 + 1280*a^4*b^2*c^12 - 1024*a^5*c^13)
*d^30)^(1/4)*(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*log(-117*(b^2*c^2 - 4*a*c^3)*sqrt(2*c*d
*x + b*d)*d^7 + 117*((b^10*c^8 - 20*a*b^8*c^9 + 160*a^2*b^6*c^10 - 640*a^3*b^4*c^11 + 1280*a^4*b^2*c^12 - 1024
*a^5*c^13)*d^30)^(1/4)) - 585*((b^10*c^8 - 20*a*b^8*c^9 + 160*a^2*b^6*c^10 - 640*a^3*b^4*c^11 + 1280*a^4*b^2*c
^12 - 1024*a^5*c^13)*d^30)^(1/4)*(-I*c^2*x^4 - 2*I*b*c*x^3 - 2*I*a*b*x - I*(b^2 + 2*a*c)*x^2 - I*a^2)*log(-117
*(b^2*c^2 - 4*a*c^3)*sqrt(2*c*d*x + b*d)*d^7 + 117*I*((b^10*c^8 - 20*a*b^8*c^9 + 160*a^2*b^6*c^10 - 640*a^3*b^
4*c^11 + 1280*a^4*b^2*c^12 - 1024*a^5*c^13)*d^30)^(1/4)) - 585*((b^10*c^8 - 20*a*b^8*c^9 + 160*a^2*b^6*c^10 -
640*a^3*b^4*c^11 + 1280*a^4*b^2*c^12 - 1024*a^5*c^13)*d^30)^(1/4)*(I*c^2*x^4 + 2*I*b*c*x^3 + 2*I*a*b*x + I*(b^
2 + 2*a*c)*x^2 + I*a^2)*log(-117*(b^2*c^2 - 4*a*c^3)*sqrt(2*c*d*x + b*d)*d^7 - 117*I*((b^10*c^8 - 20*a*b^8*c^9
+ 160*a^2*b^6*c^10 - 640*a^3*b^4*c^11 + 1280*a^4*b^2*c^12 - 1024*a^5*c^13)*d^30)^(1/4)) - 585*((b^10*c^8 - 20
*a*b^8*c^9 + 160*a^2*b^6*c^10 - 640*a^3*b^4*c^11 + 1280*a^4*b^2*c^12 - 1024*a^5*c^13)*d^30)^(1/4)*(c^2*x^4 + 2
*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*log(-117*(b^2*c^2 - 4*a*c^3)*sqrt(2*c*d*x + b*d)*d^7 - 117*((b^1
0*c^8 - 20*a*b^8*c^9 + 160*a^2*b^6*c^10 - 640*a^3*b^4*c^11 + 1280*a^4*b^2*c^12 - 1024*a^5*c^13)*d^30)^(1/4)) +
(512*c^6*d^7*x^6 + 1536*b*c^5*d^7*x^5 + 512*(7*b^2*c^4 - 13*a*c^5)*d^7*x^4 + 512*(9*b^3*c^3 - 26*a*b*c^4)*d^7
*x^3 + 3*(641*b^4*c^2 - 520*a*b^2*c^3 - 5616*a^2*c^4)*d^7*x^2 - (125*b^5*c - 5096*a*b^3*c^2 + 16848*a^2*b*c^3)
*d^7*x - (5*b^6 + 65*a*b^4*c - 2808*a^2*b^2*c^2 + 9360*a^3*c^3)*d^7)*sqrt(2*c*d*x + b*d))/(c^2*x^4 + 2*b*c*x^3
+ 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate((2*c*d*x+b*d)**(15/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((2*c*d*x+b*d)^(15/2)/(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. 753 vs. \(2 (188) = 376\).

Time = 0.31 (sec) , antiderivative size = 753, normalized size of antiderivative = 3.39 \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^3} \, dx=192 \, \sqrt {2 \, c d x + b d} b^{2} c^{2} d^{7} - 768 \, \sqrt {2 \, c d x + b d} a c^{3} d^{7} + \frac {64}{5} \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} c^{2} d^{5} - \frac {117}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{2} c^{2} d^{7} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a c^{3} d^{7}\right )} \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 ) - \frac {117}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{2} c^{2} d^{7} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a c^{3} d^{7}\right )} \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 ) - \frac {117}{4} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{2} c^{2} d^{7} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a c^{3} d^{7}\right )} \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 ) + \frac {117}{4} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{2} c^{2} d^{7} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a c^{3} d^{7}\right )} \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 ) + \frac {2 \, {\left (21 \, \sqrt {2 \, c d x + b d} b^{6} c^{2} d^{11} - 252 \, \sqrt {2 \, c d x + b d} a b^{4} c^{3} d^{11} + 1008 \, \sqrt {2 \, c d x + b d} a^{2} b^{2} c^{4} d^{11} - 1344 \, \sqrt {2 \, c d x + b d} a^{3} c^{5} d^{11} - 25 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{4} c^{2} d^{9} + 200 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} a b^{2} c^{3} d^{9} - 400 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} a^{2} c^{4} d^{9}\right )}}{{\left (b^{2} d^{2} - 4 \, a c d^{2} - {\left (2 \, c d x + b d\right )}^{2}\right )}^{2}} \]

[In]

integrate((2*c*d*x+b*d)^(15/2)/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

192*sqrt(2*c*d*x + b*d)*b^2*c^2*d^7 - 768*sqrt(2*c*d*x + b*d)*a*c^3*d^7 + 64/5*(2*c*d*x + b*d)^(5/2)*c^2*d^5 -
117/2*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*b^2*c^2*d^7 - 4*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*a*c^3*d^7)*a
rctan(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))
- 117/2*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*b^2*c^2*d^7 - 4*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*a*c^3*d^7)
*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)) - 117/4*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*b^2*c^2*d^7 - 4*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*a*c^3*d
^7)*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))
+ 117/4*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*b^2*c^2*d^7 - 4*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*a*c^3*d^7)
*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)) +
2*(21*sqrt(2*c*d*x + b*d)*b^6*c^2*d^11 - 252*sqrt(2*c*d*x + b*d)*a*b^4*c^3*d^11 + 1008*sqrt(2*c*d*x + b*d)*a^2
*b^2*c^4*d^11 - 1344*sqrt(2*c*d*x + b*d)*a^3*c^5*d^11 - 25*(2*c*d*x + b*d)^(5/2)*b^4*c^2*d^9 + 200*(2*c*d*x +
b*d)^(5/2)*a*b^2*c^3*d^9 - 400*(2*c*d*x + b*d)^(5/2)*a^2*c^4*d^9)/(b^2*d^2 - 4*a*c*d^2 - (2*c*d*x + b*d)^2)^2

Mupad [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 966, normalized size of antiderivative = 4.35 \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^3} \, dx=\frac {64\,c^2\,d^5\,{\left (b\,d+2\,c\,d\,x\right )}^{5/2}}{5}-\frac {{\left (b\,d+2\,c\,d\,x\right )}^{5/2}\,\left (800\,a^2\,c^4\,d^9-400\,a\,b^2\,c^3\,d^9+50\,b^4\,c^2\,d^9\right )+\sqrt {b\,d+2\,c\,d\,x}\,\left (2688\,a^3\,c^5\,d^{11}-2016\,a^2\,b^2\,c^4\,d^{11}+504\,a\,b^4\,c^3\,d^{11}-42\,b^6\,c^2\,d^{11}\right )}{{\left (b\,d+2\,c\,d\,x\right )}^4-{\left (b\,d+2\,c\,d\,x\right )}^2\,\left (2\,b^2\,d^2-8\,a\,c\,d^2\right )+b^4\,d^4+16\,a^2\,c^2\,d^4-8\,a\,b^2\,c\,d^4}-192\,c^2\,d^7\,\sqrt {b\,d+2\,c\,d\,x}\,\left (4\,a\,c-b^2\right )-117\,c^2\,d^{15/2}\,\mathrm {atan}\left (\frac {b^2\,\sqrt {b\,d+2\,c\,d\,x}-4\,a\,c\,\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{5/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{5/4}+c^2\,d^{15/2}\,\mathrm {atan}\left (\frac {\frac {c^2\,d^{15/2}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (56070144\,a^4\,c^8\,d^{18}-56070144\,a^3\,b^2\,c^7\,d^{18}[phone]\,a^2\,b^4\,c^6\,d^{18}-3504384\,a\,b^6\,c^5\,d^{18}+219024\,b^8\,c^4\,d^{18}\right )-\frac {117\,c^2\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{5/4}\,\left (239616\,a^3\,c^5\,d^{11}-179712\,a^2\,b^2\,c^4\,d^{11}+44928\,a\,b^4\,c^3\,d^{11}-3744\,b^6\,c^2\,d^{11}\right )}{2}\right )\,{\left (b^2-4\,a\,c\right )}^{5/4}\,117{}\mathrm {i}}{2}+\frac {c^2\,d^{15/2}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (56070144\,a^4\,c^8\,d^{18}-56070144\,a^3\,b^2\,c^7\,d^{18}[phone]\,a^2\,b^4\,c^6\,d^{18}-3504384\,a\,b^6\,c^5\,d^{18}+219024\,b^8\,c^4\,d^{18}\right )+\frac {117\,c^2\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{5/4}\,\left (239616\,a^3\,c^5\,d^{11}-179712\,a^2\,b^2\,c^4\,d^{11}+44928\,a\,b^4\,c^3\,d^{11}-3744\,b^6\,c^2\,d^{11}\right )}{2}\right )\,{\left (b^2-4\,a\,c\right )}^{5/4}\,117{}\mathrm {i}}{2}}{\frac {117\,c^2\,d^{15/2}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (56070144\,a^4\,c^8\,d^{18}-56070144\,a^3\,b^2\,c^7\,d^{18}[phone]\,a^2\,b^4\,c^6\,d^{18}-3504384\,a\,b^6\,c^5\,d^{18}+219024\,b^8\,c^4\,d^{18}\right )-\frac {117\,c^2\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{5/4}\,\left (239616\,a^3\,c^5\,d^{11}-179712\,a^2\,b^2\,c^4\,d^{11}+44928\,a\,b^4\,c^3\,d^{11}-3744\,b^6\,c^2\,d^{11}\right )}{2}\right )\,{\left (b^2-4\,a\,c\right )}^{5/4}}{2}-\frac {117\,c^2\,d^{15/2}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (56070144\,a^4\,c^8\,d^{18}-56070144\,a^3\,b^2\,c^7\,d^{18}[phone]\,a^2\,b^4\,c^6\,d^{18}-3504384\,a\,b^6\,c^5\,d^{18}+219024\,b^8\,c^4\,d^{18}\right )+\frac {117\,c^2\,d^{15/2}\,{\left (b^2-4\,a\,c\right )}^{5/4}\,\left (239616\,a^3\,c^5\,d^{11}-179712\,a^2\,b^2\,c^4\,d^{11}+44928\,a\,b^4\,c^3\,d^{11}-3744\,b^6\,c^2\,d^{11}\right )}{2}\right )\,{\left (b^2-4\,a\,c\right )}^{5/4}}{2}}\right )\,{\left (b^2-4\,a\,c\right )}^{5/4}\,117{}\mathrm {i} \]

[In]

int((b*d + 2*c*d*x)^(15/2)/(a + b*x + c*x^2)^3,x)

[Out]

(64*c^2*d^5*(b*d + 2*c*d*x)^(5/2))/5 - ((b*d + 2*c*d*x)^(5/2)*(800*a^2*c^4*d^9 + 50*b^4*c^2*d^9 - 400*a*b^2*c^
3*d^9) + (b*d + 2*c*d*x)^(1/2)*(2688*a^3*c^5*d^11 - 42*b^6*c^2*d^11 + 504*a*b^4*c^3*d^11 - 2016*a^2*b^2*c^4*d^
11))/((b*d + 2*c*d*x)^4 - (b*d + 2*c*d*x)^2*(2*b^2*d^2 - 8*a*c*d^2) + b^4*d^4 + 16*a^2*c^2*d^4 - 8*a*b^2*c*d^4
) - 192*c^2*d^7*(b*d + 2*c*d*x)^(1/2)*(4*a*c - b^2) + c^2*d^(15/2)*atan(((c^2*d^(15/2)*((b*d + 2*c*d*x)^(1/2)*
(56070144*a^4*c^8*d^18 + 219024*b^8*c^4*d^18 - 3504384*a*b^6*c^5*d^18 + 21026304*a^2*b^4*c^6*d^18 - 56070144*a
^3*b^2*c^7*d^18) - (117*c^2*d^(15/2)*(b^2 - 4*a*c)^(5/4)*(239616*a^3*c^5*d^11 - 3744*b^6*c^2*d^11 + 44928*a*b^
4*c^3*d^11 - 179712*a^2*b^2*c^4*d^11))/2)*(b^2 - 4*a*c)^(5/4)*117i)/2 + (c^2*d^(15/2)*((b*d + 2*c*d*x)^(1/2)*(
56070144*a^4*c^8*d^18 + 219024*b^8*c^4*d^18 - 3504384*a*b^6*c^5*d^18 + 21026304*a^2*b^4*c^6*d^18 - 56070144*a^
3*b^2*c^7*d^18) + (117*c^2*d^(15/2)*(b^2 - 4*a*c)^(5/4)*(239616*a^3*c^5*d^11 - 3744*b^6*c^2*d^11 + 44928*a*b^4
*c^3*d^11 - 179712*a^2*b^2*c^4*d^11))/2)*(b^2 - 4*a*c)^(5/4)*117i)/2)/((117*c^2*d^(15/2)*((b*d + 2*c*d*x)^(1/2
)*(56070144*a^4*c^8*d^18 + 219024*b^8*c^4*d^18 - 3504384*a*b^6*c^5*d^18 + 21026304*a^2*b^4*c^6*d^18 - 56070144
*a^3*b^2*c^7*d^18) - (117*c^2*d^(15/2)*(b^2 - 4*a*c)^(5/4)*(239616*a^3*c^5*d^11 - 3744*b^6*c^2*d^11 + 44928*a*
b^4*c^3*d^11 - 179712*a^2*b^2*c^4*d^11))/2)*(b^2 - 4*a*c)^(5/4))/2 - (117*c^2*d^(15/2)*((b*d + 2*c*d*x)^(1/2)*
(56070144*a^4*c^8*d^18 + 219024*b^8*c^4*d^18 - 3504384*a*b^6*c^5*d^18 + 21026304*a^2*b^4*c^6*d^18 - 56070144*a
^3*b^2*c^7*d^18) + (117*c^2*d^(15/2)*(b^2 - 4*a*c)^(5/4)*(239616*a^3*c^5*d^11 - 3744*b^6*c^2*d^11 + 44928*a*b^
4*c^3*d^11 - 179712*a^2*b^2*c^4*d^11))/2)*(b^2 - 4*a*c)^(5/4))/2))*(b^2 - 4*a*c)^(5/4)*117i - 117*c^2*d^(15/2)
*atan((b^2*(b*d + 2*c*d*x)^(1/2) - 4*a*c*(b*d + 2*c*d*x)^(1/2))/(d^(1/2)*(b^2 - 4*a*c)^(5/4)))*(b^2 - 4*a*c)^(
5/4)