Optimal. Leaf size=334 \[ \frac{c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{6 e^8 (d+e x)^6}-\frac{3 c^2 \left (a B e^2-2 A c d e+7 B c d^2\right )}{4 e^8 (d+e x)^4}+\frac{c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{5 e^8 (d+e x)^5}+\frac{3 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{7 e^8 (d+e x)^7}-\frac{\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{8 e^8 (d+e x)^8}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{9 e^8 (d+e x)^9}+\frac{c^3 (7 B d-A e)}{3 e^8 (d+e x)^3}-\frac{B c^3}{2 e^8 (d+e x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.26433, antiderivative size = 334, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {772} \[ \frac{c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{6 e^8 (d+e x)^6}-\frac{3 c^2 \left (a B e^2-2 A c d e+7 B c d^2\right )}{4 e^8 (d+e x)^4}+\frac{c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{5 e^8 (d+e x)^5}+\frac{3 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{7 e^8 (d+e x)^7}-\frac{\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{8 e^8 (d+e x)^8}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{9 e^8 (d+e x)^9}+\frac{c^3 (7 B d-A e)}{3 e^8 (d+e x)^3}-\frac{B c^3}{2 e^8 (d+e x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 772
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{10}} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )^3}{e^7 (d+e x)^{10}}+\frac{\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{e^7 (d+e x)^9}+\frac{3 c \left (c d^2+a e^2\right ) \left (-7 B c d^3+5 A c d^2 e-3 a B d e^2+a A e^3\right )}{e^7 (d+e x)^8}-\frac{c \left (-35 B c^2 d^4+20 A c^2 d^3 e-30 a B c d^2 e^2+12 a A c d e^3-3 a^2 B e^4\right )}{e^7 (d+e x)^7}+\frac{c^2 \left (-35 B c d^3+15 A c d^2 e-15 a B d e^2+3 a A e^3\right )}{e^7 (d+e x)^6}-\frac{3 c^2 \left (-7 B c d^2+2 A c d e-a B e^2\right )}{e^7 (d+e x)^5}+\frac{c^3 (-7 B d+A e)}{e^7 (d+e x)^4}+\frac{B c^3}{e^7 (d+e x)^3}\right ) \, dx\\ &=\frac{(B d-A e) \left (c d^2+a e^2\right )^3}{9 e^8 (d+e x)^9}-\frac{\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{8 e^8 (d+e x)^8}+\frac{3 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right )}{7 e^8 (d+e x)^7}+\frac{c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right )}{6 e^8 (d+e x)^6}+\frac{c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right )}{5 e^8 (d+e x)^5}-\frac{3 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right )}{4 e^8 (d+e x)^4}+\frac{c^3 (7 B d-A e)}{3 e^8 (d+e x)^3}-\frac{B c^3}{2 e^8 (d+e x)^2}\\ \end{align*}
Mathematica [A] time = 0.182331, size = 359, normalized size = 1.07 \[ -\frac{2 A e \left (15 a^2 c e^4 \left (d^2+9 d e x+36 e^2 x^2\right )+140 a^3 e^6+6 a c^2 e^2 \left (36 d^2 e^2 x^2+9 d^3 e x+d^4+84 d e^3 x^3+126 e^4 x^4\right )+5 c^3 \left (36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+9 d^5 e x+d^6+126 d e^5 x^5+84 e^6 x^6\right )\right )+5 B \left (3 a^2 c e^4 \left (9 d^2 e x+d^3+36 d e^2 x^2+84 e^3 x^3\right )+7 a^3 e^6 (d+9 e x)+3 a c^2 e^2 \left (36 d^3 e^2 x^2+84 d^2 e^3 x^3+9 d^4 e x+d^5+126 d e^4 x^4+126 e^5 x^5\right )+7 c^3 \left (36 d^5 e^2 x^2+84 d^4 e^3 x^3+126 d^3 e^4 x^4+126 d^2 e^5 x^5+9 d^6 e x+d^7+84 d e^6 x^6+36 e^7 x^7\right )\right )}{2520 e^8 (d+e x)^9} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.009, size = 449, normalized size = 1.3 \begin{align*} -{\frac{3\,c \left ( A{a}^{2}{e}^{5}+6\,A{d}^{2}ac{e}^{3}+5\,A{d}^{4}{c}^{2}e-3\,B{a}^{2}d{e}^{4}-10\,aBc{d}^{3}{e}^{2}-7\,B{c}^{2}{d}^{5} \right ) }{7\,{e}^{8} \left ( ex+d \right ) ^{7}}}-{\frac{B{c}^{3}}{2\,{e}^{8} \left ( ex+d \right ) ^{2}}}-{\frac{{c}^{3} \left ( Ae-7\,Bd \right ) }{3\,{e}^{8} \left ( ex+d \right ) ^{3}}}+{\frac{3\,{c}^{2} \left ( 2\,Acde-aB{e}^{2}-7\,Bc{d}^{2} \right ) }{4\,{e}^{8} \left ( ex+d \right ) ^{4}}}-{\frac{-6\,A{a}^{2}cd{e}^{5}-12\,A{d}^{3}a{c}^{2}{e}^{3}-6\,A{d}^{5}{c}^{3}e+B{a}^{3}{e}^{6}+9\,B{a}^{2}c{d}^{2}{e}^{4}+15\,Ba{c}^{2}{d}^{4}{e}^{2}+7\,B{c}^{3}{d}^{6}}{8\,{e}^{8} \left ( ex+d \right ) ^{8}}}-{\frac{{c}^{2} \left ( 3\,aA{e}^{3}+15\,Ac{d}^{2}e-15\,aBd{e}^{2}-35\,Bc{d}^{3} \right ) }{5\,{e}^{8} \left ( ex+d \right ) ^{5}}}-{\frac{A{a}^{3}{e}^{7}+3\,A{d}^{2}{a}^{2}c{e}^{5}+3\,A{d}^{4}a{c}^{2}{e}^{3}+A{d}^{6}{c}^{3}e-B{a}^{3}d{e}^{6}-3\,B{a}^{2}c{d}^{3}{e}^{4}-3\,Ba{c}^{2}{d}^{5}{e}^{2}-B{c}^{3}{d}^{7}}{9\,{e}^{8} \left ( ex+d \right ) ^{9}}}+{\frac{c \left ( 12\,Adac{e}^{3}+20\,A{c}^{2}{d}^{3}e-3\,B{e}^{4}{a}^{2}-30\,Bac{d}^{2}{e}^{2}-35\,B{c}^{2}{d}^{4} \right ) }{6\,{e}^{8} \left ( ex+d \right ) ^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.16437, size = 737, normalized size = 2.21 \begin{align*} -\frac{1260 \, B c^{3} e^{7} x^{7} + 35 \, B c^{3} d^{7} + 10 \, A c^{3} d^{6} e + 15 \, B a c^{2} d^{5} e^{2} + 12 \, A a c^{2} d^{4} e^{3} + 15 \, B a^{2} c d^{3} e^{4} + 30 \, A a^{2} c d^{2} e^{5} + 35 \, B a^{3} d e^{6} + 280 \, A a^{3} e^{7} + 420 \,{\left (7 \, B c^{3} d e^{6} + 2 \, A c^{3} e^{7}\right )} x^{6} + 630 \,{\left (7 \, B c^{3} d^{2} e^{5} + 2 \, A c^{3} d e^{6} + 3 \, B a c^{2} e^{7}\right )} x^{5} + 126 \,{\left (35 \, B c^{3} d^{3} e^{4} + 10 \, A c^{3} d^{2} e^{5} + 15 \, B a c^{2} d e^{6} + 12 \, A a c^{2} e^{7}\right )} x^{4} + 84 \,{\left (35 \, B c^{3} d^{4} e^{3} + 10 \, A c^{3} d^{3} e^{4} + 15 \, B a c^{2} d^{2} e^{5} + 12 \, A a c^{2} d e^{6} + 15 \, B a^{2} c e^{7}\right )} x^{3} + 36 \,{\left (35 \, B c^{3} d^{5} e^{2} + 10 \, A c^{3} d^{4} e^{3} + 15 \, B a c^{2} d^{3} e^{4} + 12 \, A a c^{2} d^{2} e^{5} + 15 \, B a^{2} c d e^{6} + 30 \, A a^{2} c e^{7}\right )} x^{2} + 9 \,{\left (35 \, B c^{3} d^{6} e + 10 \, A c^{3} d^{5} e^{2} + 15 \, B a c^{2} d^{4} e^{3} + 12 \, A a c^{2} d^{3} e^{4} + 15 \, B a^{2} c d^{2} e^{5} + 30 \, A a^{2} c d e^{6} + 35 \, B a^{3} e^{7}\right )} x}{2520 \,{\left (e^{17} x^{9} + 9 \, d e^{16} x^{8} + 36 \, d^{2} e^{15} x^{7} + 84 \, d^{3} e^{14} x^{6} + 126 \, d^{4} e^{13} x^{5} + 126 \, d^{5} e^{12} x^{4} + 84 \, d^{6} e^{11} x^{3} + 36 \, d^{7} e^{10} x^{2} + 9 \, d^{8} e^{9} x + d^{9} e^{8}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.86456, size = 1203, normalized size = 3.6 \begin{align*} -\frac{1260 \, B c^{3} e^{7} x^{7} + 35 \, B c^{3} d^{7} + 10 \, A c^{3} d^{6} e + 15 \, B a c^{2} d^{5} e^{2} + 12 \, A a c^{2} d^{4} e^{3} + 15 \, B a^{2} c d^{3} e^{4} + 30 \, A a^{2} c d^{2} e^{5} + 35 \, B a^{3} d e^{6} + 280 \, A a^{3} e^{7} + 420 \,{\left (7 \, B c^{3} d e^{6} + 2 \, A c^{3} e^{7}\right )} x^{6} + 630 \,{\left (7 \, B c^{3} d^{2} e^{5} + 2 \, A c^{3} d e^{6} + 3 \, B a c^{2} e^{7}\right )} x^{5} + 126 \,{\left (35 \, B c^{3} d^{3} e^{4} + 10 \, A c^{3} d^{2} e^{5} + 15 \, B a c^{2} d e^{6} + 12 \, A a c^{2} e^{7}\right )} x^{4} + 84 \,{\left (35 \, B c^{3} d^{4} e^{3} + 10 \, A c^{3} d^{3} e^{4} + 15 \, B a c^{2} d^{2} e^{5} + 12 \, A a c^{2} d e^{6} + 15 \, B a^{2} c e^{7}\right )} x^{3} + 36 \,{\left (35 \, B c^{3} d^{5} e^{2} + 10 \, A c^{3} d^{4} e^{3} + 15 \, B a c^{2} d^{3} e^{4} + 12 \, A a c^{2} d^{2} e^{5} + 15 \, B a^{2} c d e^{6} + 30 \, A a^{2} c e^{7}\right )} x^{2} + 9 \,{\left (35 \, B c^{3} d^{6} e + 10 \, A c^{3} d^{5} e^{2} + 15 \, B a c^{2} d^{4} e^{3} + 12 \, A a c^{2} d^{3} e^{4} + 15 \, B a^{2} c d^{2} e^{5} + 30 \, A a^{2} c d e^{6} + 35 \, B a^{3} e^{7}\right )} x}{2520 \,{\left (e^{17} x^{9} + 9 \, d e^{16} x^{8} + 36 \, d^{2} e^{15} x^{7} + 84 \, d^{3} e^{14} x^{6} + 126 \, d^{4} e^{13} x^{5} + 126 \, d^{5} e^{12} x^{4} + 84 \, d^{6} e^{11} x^{3} + 36 \, d^{7} e^{10} x^{2} + 9 \, d^{8} e^{9} x + d^{9} e^{8}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.20746, size = 617, normalized size = 1.85 \begin{align*} -\frac{{\left (1260 \, B c^{3} x^{7} e^{7} + 2940 \, B c^{3} d x^{6} e^{6} + 4410 \, B c^{3} d^{2} x^{5} e^{5} + 4410 \, B c^{3} d^{3} x^{4} e^{4} + 2940 \, B c^{3} d^{4} x^{3} e^{3} + 1260 \, B c^{3} d^{5} x^{2} e^{2} + 315 \, B c^{3} d^{6} x e + 35 \, B c^{3} d^{7} + 840 \, A c^{3} x^{6} e^{7} + 1260 \, A c^{3} d x^{5} e^{6} + 1260 \, A c^{3} d^{2} x^{4} e^{5} + 840 \, A c^{3} d^{3} x^{3} e^{4} + 360 \, A c^{3} d^{4} x^{2} e^{3} + 90 \, A c^{3} d^{5} x e^{2} + 10 \, A c^{3} d^{6} e + 1890 \, B a c^{2} x^{5} e^{7} + 1890 \, B a c^{2} d x^{4} e^{6} + 1260 \, B a c^{2} d^{2} x^{3} e^{5} + 540 \, B a c^{2} d^{3} x^{2} e^{4} + 135 \, B a c^{2} d^{4} x e^{3} + 15 \, B a c^{2} d^{5} e^{2} + 1512 \, A a c^{2} x^{4} e^{7} + 1008 \, A a c^{2} d x^{3} e^{6} + 432 \, A a c^{2} d^{2} x^{2} e^{5} + 108 \, A a c^{2} d^{3} x e^{4} + 12 \, A a c^{2} d^{4} e^{3} + 1260 \, B a^{2} c x^{3} e^{7} + 540 \, B a^{2} c d x^{2} e^{6} + 135 \, B a^{2} c d^{2} x e^{5} + 15 \, B a^{2} c d^{3} e^{4} + 1080 \, A a^{2} c x^{2} e^{7} + 270 \, A a^{2} c d x e^{6} + 30 \, A a^{2} c d^{2} e^{5} + 315 \, B a^{3} x e^{7} + 35 \, B a^{3} d e^{6} + 280 \, A a^{3} e^{7}\right )} e^{\left (-8\right )}}{2520 \,{\left (x e + d\right )}^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]