Optimal. Leaf size=234 \[ -\frac{3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7 (d+e x)^5}+\frac{(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{6 e^7 (d+e x)^6}-\frac{3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{7 e^7 (d+e x)^7}+\frac{3 c^2 (2 c d-b e)}{4 e^7 (d+e x)^4}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{8 e^7 (d+e x)^8}-\frac{d^3 (c d-b e)^3}{9 e^7 (d+e x)^9}-\frac{c^3}{3 e^7 (d+e x)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.156602, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {698} \[ -\frac{3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7 (d+e x)^5}+\frac{(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{6 e^7 (d+e x)^6}-\frac{3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{7 e^7 (d+e x)^7}+\frac{3 c^2 (2 c d-b e)}{4 e^7 (d+e x)^4}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{8 e^7 (d+e x)^8}-\frac{d^3 (c d-b e)^3}{9 e^7 (d+e x)^9}-\frac{c^3}{3 e^7 (d+e x)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 698
Rubi steps
\begin{align*} \int \frac{\left (b x+c x^2\right )^3}{(d+e x)^{10}} \, dx &=\int \left (\frac{d^3 (c d-b e)^3}{e^6 (d+e x)^{10}}-\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 (d+e x)^9}+\frac{3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^6 (d+e x)^8}+\frac{(2 c d-b e) \left (-10 c^2 d^2+10 b c d e-b^2 e^2\right )}{e^6 (d+e x)^7}+\frac{3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^6 (d+e x)^6}-\frac{3 c^2 (2 c d-b e)}{e^6 (d+e x)^5}+\frac{c^3}{e^6 (d+e x)^4}\right ) \, dx\\ &=-\frac{d^3 (c d-b e)^3}{9 e^7 (d+e x)^9}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{8 e^7 (d+e x)^8}-\frac{3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{7 e^7 (d+e x)^7}+\frac{(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right )}{6 e^7 (d+e x)^6}-\frac{3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{5 e^7 (d+e x)^5}+\frac{3 c^2 (2 c d-b e)}{4 e^7 (d+e x)^4}-\frac{c^3}{3 e^7 (d+e x)^3}\\ \end{align*}
Mathematica [A] time = 0.0746255, size = 222, normalized size = 0.95 \[ -\frac{12 b^2 c e^2 \left (36 d^2 e^2 x^2+9 d^3 e x+d^4+84 d e^3 x^3+126 e^4 x^4\right )+5 b^3 e^3 \left (9 d^2 e x+d^3+36 d e^2 x^2+84 e^3 x^3\right )+15 b c^2 e \left (36 d^3 e^2 x^2+84 d^2 e^3 x^3+9 d^4 e x+d^5+126 d e^4 x^4+126 e^5 x^5\right )+10 c^3 \left (36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+9 d^5 e x+d^6+126 d e^5 x^5+84 e^6 x^6\right )}{2520 e^7 (d+e x)^9} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.051, size = 274, normalized size = 1.2 \begin{align*}{\frac{{d}^{3} \left ({b}^{3}{e}^{3}-3\,{b}^{2}cd{e}^{2}+3\,b{c}^{2}{d}^{2}e-{c}^{3}{d}^{3} \right ) }{9\,{e}^{7} \left ( ex+d \right ) ^{9}}}+{\frac{3\,d \left ({b}^{3}{e}^{3}-6\,{b}^{2}cd{e}^{2}+10\,b{c}^{2}{d}^{2}e-5\,{c}^{3}{d}^{3} \right ) }{7\,{e}^{7} \left ( ex+d \right ) ^{7}}}-{\frac{3\,c \left ({b}^{2}{e}^{2}-5\,bcde+5\,{c}^{2}{d}^{2} \right ) }{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}-{\frac{{b}^{3}{e}^{3}-12\,{b}^{2}cd{e}^{2}+30\,b{c}^{2}{d}^{2}e-20\,{c}^{3}{d}^{3}}{6\,{e}^{7} \left ( ex+d \right ) ^{6}}}-{\frac{3\,{d}^{2} \left ({b}^{3}{e}^{3}-4\,{b}^{2}cd{e}^{2}+5\,b{c}^{2}{d}^{2}e-2\,{c}^{3}{d}^{3} \right ) }{8\,{e}^{7} \left ( ex+d \right ) ^{8}}}-{\frac{{c}^{3}}{3\,{e}^{7} \left ( ex+d \right ) ^{3}}}-{\frac{3\,{c}^{2} \left ( be-2\,cd \right ) }{4\,{e}^{7} \left ( ex+d \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.23416, size = 487, normalized size = 2.08 \begin{align*} -\frac{840 \, c^{3} e^{6} x^{6} + 10 \, c^{3} d^{6} + 15 \, b c^{2} d^{5} e + 12 \, b^{2} c d^{4} e^{2} + 5 \, b^{3} d^{3} e^{3} + 630 \,{\left (2 \, c^{3} d e^{5} + 3 \, b c^{2} e^{6}\right )} x^{5} + 126 \,{\left (10 \, c^{3} d^{2} e^{4} + 15 \, b c^{2} d e^{5} + 12 \, b^{2} c e^{6}\right )} x^{4} + 84 \,{\left (10 \, c^{3} d^{3} e^{3} + 15 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} + 5 \, b^{3} e^{6}\right )} x^{3} + 36 \,{\left (10 \, c^{3} d^{4} e^{2} + 15 \, b c^{2} d^{3} e^{3} + 12 \, b^{2} c d^{2} e^{4} + 5 \, b^{3} d e^{5}\right )} x^{2} + 9 \,{\left (10 \, c^{3} d^{5} e + 15 \, b c^{2} d^{4} e^{2} + 12 \, b^{2} c d^{3} e^{3} + 5 \, b^{3} d^{2} e^{4}\right )} x}{2520 \,{\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.60613, size = 776, normalized size = 3.32 \begin{align*} -\frac{840 \, c^{3} e^{6} x^{6} + 10 \, c^{3} d^{6} + 15 \, b c^{2} d^{5} e + 12 \, b^{2} c d^{4} e^{2} + 5 \, b^{3} d^{3} e^{3} + 630 \,{\left (2 \, c^{3} d e^{5} + 3 \, b c^{2} e^{6}\right )} x^{5} + 126 \,{\left (10 \, c^{3} d^{2} e^{4} + 15 \, b c^{2} d e^{5} + 12 \, b^{2} c e^{6}\right )} x^{4} + 84 \,{\left (10 \, c^{3} d^{3} e^{3} + 15 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} + 5 \, b^{3} e^{6}\right )} x^{3} + 36 \,{\left (10 \, c^{3} d^{4} e^{2} + 15 \, b c^{2} d^{3} e^{3} + 12 \, b^{2} c d^{2} e^{4} + 5 \, b^{3} d e^{5}\right )} x^{2} + 9 \,{\left (10 \, c^{3} d^{5} e + 15 \, b c^{2} d^{4} e^{2} + 12 \, b^{2} c d^{3} e^{3} + 5 \, b^{3} d^{2} e^{4}\right )} x}{2520 \,{\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} e^{7}\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 = 1.25734, size = 362, normalized size = 1.55 \begin{align*} -\frac{{\left (840 \, c^{3} x^{6} e^{6} + 1260 \, c^{3} d x^{5} e^{5} + 1260 \, c^{3} d^{2} x^{4} e^{4} + 840 \, c^{3} d^{3} x^{3} e^{3} + 360 \, c^{3} d^{4} x^{2} e^{2} + 90 \, c^{3} d^{5} x e + 10 \, c^{3} d^{6} + 1890 \, b c^{2} x^{5} e^{6} + 1890 \, b c^{2} d x^{4} e^{5} + 1260 \, b c^{2} d^{2} x^{3} e^{4} + 540 \, b c^{2} d^{3} x^{2} e^{3} + 135 \, b c^{2} d^{4} x e^{2} + 15 \, b c^{2} d^{5} e + 1512 \, b^{2} c x^{4} e^{6} + 1008 \, b^{2} c d x^{3} e^{5} + 432 \, b^{2} c d^{2} x^{2} e^{4} + 108 \, b^{2} c d^{3} x e^{3} + 12 \, b^{2} c d^{4} e^{2} + 420 \, b^{3} x^{3} e^{6} + 180 \, b^{3} d x^{2} e^{5} + 45 \, b^{3} d^{2} x e^{4} + 5 \, b^{3} d^{3} e^{3}\right )} e^{\left (-7\right )}}{2520 \,{\left (x e + d\right )}^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]