Optimal. Leaf size=422 \[ -\frac{c x^2 \left (A c e (4 c d-3 b e)-B \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{2 e^6}+\frac{x \left (A c e \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )-B \left (12 b^2 c d e^2-b^3 e^3-30 b c^2 d^2 e+20 c^3 d^3\right )\right )}{e^7}-\frac{3 d (c d-b e) \left (A e \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-B d \left (2 b^2 e^2-8 b c d e+7 c^2 d^2\right )\right )}{e^8 (d+e x)}+\frac{\log (d+e x) \left (B d \left (30 b^2 c d e^2-4 b^3 e^3-60 b c^2 d^2 e+35 c^3 d^3\right )-A e \left (12 b^2 c d e^2-b^3 e^3-30 b c^2 d^2 e+20 c^3 d^3\right )\right )}{e^8}-\frac{c^2 x^3 (-A c e-3 b B e+4 B c d)}{3 e^5}-\frac{d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{2 e^8 (d+e x)^2}+\frac{d^3 (B d-A e) (c d-b e)^3}{3 e^8 (d+e x)^3}+\frac{B c^3 x^4}{4 e^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.635539, antiderivative size = 422, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ -\frac{c x^2 \left (A c e (4 c d-3 b e)-B \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{2 e^6}+\frac{x \left (A c e \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )-B \left (12 b^2 c d e^2-b^3 e^3-30 b c^2 d^2 e+20 c^3 d^3\right )\right )}{e^7}-\frac{3 d (c d-b e) \left (A e \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-B d \left (2 b^2 e^2-8 b c d e+7 c^2 d^2\right )\right )}{e^8 (d+e x)}+\frac{\log (d+e x) \left (B d \left (30 b^2 c d e^2-4 b^3 e^3-60 b c^2 d^2 e+35 c^3 d^3\right )-A e \left (12 b^2 c d e^2-b^3 e^3-30 b c^2 d^2 e+20 c^3 d^3\right )\right )}{e^8}-\frac{c^2 x^3 (-A c e-3 b B e+4 B c d)}{3 e^5}-\frac{d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{2 e^8 (d+e x)^2}+\frac{d^3 (B d-A e) (c d-b e)^3}{3 e^8 (d+e x)^3}+\frac{B c^3 x^4}{4 e^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 771
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^3}{(d+e x)^4} \, dx &=\int \left (\frac{A c e \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )-B \left (20 c^3 d^3-30 b c^2 d^2 e+12 b^2 c d e^2-b^3 e^3\right )}{e^7}+\frac{c \left (-A c e (4 c d-3 b e)+B \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) x}{e^6}+\frac{c^2 (-4 B c d+3 b B e+A c e) x^2}{e^5}+\frac{B c^3 x^3}{e^4}-\frac{d^3 (B d-A e) (c d-b e)^3}{e^7 (d+e x)^4}+\frac{d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{e^7 (d+e x)^3}+\frac{3 d (c d-b e) \left (A e \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )-B d \left (7 c^2 d^2-8 b c d e+2 b^2 e^2\right )\right )}{e^7 (d+e x)^2}+\frac{B d \left (35 c^3 d^3-60 b c^2 d^2 e+30 b^2 c d e^2-4 b^3 e^3\right )-A e \left (20 c^3 d^3-30 b c^2 d^2 e+12 b^2 c d e^2-b^3 e^3\right )}{e^7 (d+e x)}\right ) \, dx\\ &=\frac{\left (A c e \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )-B \left (20 c^3 d^3-30 b c^2 d^2 e+12 b^2 c d e^2-b^3 e^3\right )\right ) x}{e^7}-\frac{c \left (A c e (4 c d-3 b e)-B \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) x^2}{2 e^6}-\frac{c^2 (4 B c d-3 b B e-A c e) x^3}{3 e^5}+\frac{B c^3 x^4}{4 e^4}+\frac{d^3 (B d-A e) (c d-b e)^3}{3 e^8 (d+e x)^3}-\frac{d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{2 e^8 (d+e x)^2}-\frac{3 d (c d-b e) \left (A e \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )-B d \left (7 c^2 d^2-8 b c d e+2 b^2 e^2\right )\right )}{e^8 (d+e x)}+\frac{\left (B d \left (35 c^3 d^3-60 b c^2 d^2 e+30 b^2 c d e^2-4 b^3 e^3\right )-A e \left (20 c^3 d^3-30 b c^2 d^2 e+12 b^2 c d e^2-b^3 e^3\right )\right ) \log (d+e x)}{e^8}\\ \end{align*}
Mathematica [A] time = 0.19132, size = 400, normalized size = 0.95 \[ \frac{-6 c e^2 x^2 \left (A c e (4 c d-3 b e)+B \left (-3 b^2 e^2+12 b c d e-10 c^2 d^2\right )\right )+12 e x \left (A c e \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )+B \left (-12 b^2 c d e^2+b^3 e^3+30 b c^2 d^2 e-20 c^3 d^3\right )\right )+\frac{36 d (c d-b e) \left (B d \left (2 b^2 e^2-8 b c d e+7 c^2 d^2\right )-A e \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )\right )}{d+e x}+12 \log (d+e x) \left (A e \left (-12 b^2 c d e^2+b^3 e^3+30 b c^2 d^2 e-20 c^3 d^3\right )+B d \left (30 b^2 c d e^2-4 b^3 e^3-60 b c^2 d^2 e+35 c^3 d^3\right )\right )+4 c^2 e^3 x^3 (A c e+3 b B e-4 B c d)-\frac{6 d^2 (c d-b e)^2 (3 A e (b e-2 c d)+B d (7 c d-4 b e))}{(d+e x)^2}+\frac{4 d^3 (B d-A e) (c d-b e)^3}{(d+e x)^3}+3 B c^3 e^4 x^4}{12 e^8} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.019, size = 807, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.11029, size = 757, normalized size = 1.79 \begin{align*} \frac{107 \, B c^{3} d^{7} + 11 \, A b^{3} d^{3} e^{4} - 74 \,{\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 141 \,{\left (B b^{2} c + A b c^{2}\right )} d^{5} e^{2} - 26 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} d^{4} e^{3} + 18 \,{\left (7 \, B c^{3} d^{5} e^{2} + A b^{3} d e^{6} - 5 \,{\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 10 \,{\left (B b^{2} c + A b c^{2}\right )} d^{3} e^{4} - 2 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2} e^{5}\right )} x^{2} + 3 \,{\left (77 \, B c^{3} d^{6} e + 9 \, A b^{3} d^{2} e^{5} - 54 \,{\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 105 \,{\left (B b^{2} c + A b c^{2}\right )} d^{4} e^{3} - 20 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} d^{3} e^{4}\right )} x}{6 \,{\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} + \frac{3 \, B c^{3} e^{3} x^{4} - 4 \,{\left (4 \, B c^{3} d e^{2} -{\left (3 \, B b c^{2} + A c^{3}\right )} e^{3}\right )} x^{3} + 6 \,{\left (10 \, B c^{3} d^{2} e - 4 \,{\left (3 \, B b c^{2} + A c^{3}\right )} d e^{2} + 3 \,{\left (B b^{2} c + A b c^{2}\right )} e^{3}\right )} x^{2} - 12 \,{\left (20 \, B c^{3} d^{3} - 10 \,{\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e + 12 \,{\left (B b^{2} c + A b c^{2}\right )} d e^{2} -{\left (B b^{3} + 3 \, A b^{2} c\right )} e^{3}\right )} x}{12 \, e^{7}} + \frac{{\left (35 \, B c^{3} d^{4} + A b^{3} e^{4} - 20 \,{\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e + 30 \,{\left (B b^{2} c + A b c^{2}\right )} d^{2} e^{2} - 4 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} d e^{3}\right )} \log \left (e x + d\right )}{e^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.58754, size = 1920, normalized size = 4.55 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 62.3235, size = 688, normalized size = 1.63 \begin{align*} \frac{B c^{3} x^{4}}{4 e^{4}} - \frac{- 11 A b^{3} d^{3} e^{4} + 78 A b^{2} c d^{4} e^{3} - 141 A b c^{2} d^{5} e^{2} + 74 A c^{3} d^{6} e + 26 B b^{3} d^{4} e^{3} - 141 B b^{2} c d^{5} e^{2} + 222 B b c^{2} d^{6} e - 107 B c^{3} d^{7} + x^{2} \left (- 18 A b^{3} d e^{6} + 108 A b^{2} c d^{2} e^{5} - 180 A b c^{2} d^{3} e^{4} + 90 A c^{3} d^{4} e^{3} + 36 B b^{3} d^{2} e^{5} - 180 B b^{2} c d^{3} e^{4} + 270 B b c^{2} d^{4} e^{3} - 126 B c^{3} d^{5} e^{2}\right ) + x \left (- 27 A b^{3} d^{2} e^{5} + 180 A b^{2} c d^{3} e^{4} - 315 A b c^{2} d^{4} e^{3} + 162 A c^{3} d^{5} e^{2} + 60 B b^{3} d^{3} e^{4} - 315 B b^{2} c d^{4} e^{3} + 486 B b c^{2} d^{5} e^{2} - 231 B c^{3} d^{6} e\right )}{6 d^{3} e^{8} + 18 d^{2} e^{9} x + 18 d e^{10} x^{2} + 6 e^{11} x^{3}} + \frac{x^{3} \left (A c^{3} e + 3 B b c^{2} e - 4 B c^{3} d\right )}{3 e^{5}} + \frac{x^{2} \left (3 A b c^{2} e^{2} - 4 A c^{3} d e + 3 B b^{2} c e^{2} - 12 B b c^{2} d e + 10 B c^{3} d^{2}\right )}{2 e^{6}} + \frac{x \left (3 A b^{2} c e^{3} - 12 A b c^{2} d e^{2} + 10 A c^{3} d^{2} e + B b^{3} e^{3} - 12 B b^{2} c d e^{2} + 30 B b c^{2} d^{2} e - 20 B c^{3} d^{3}\right )}{e^{7}} - \frac{\left (- A b^{3} e^{4} + 12 A b^{2} c d e^{3} - 30 A b c^{2} d^{2} e^{2} + 20 A c^{3} d^{3} e + 4 B b^{3} d e^{3} - 30 B b^{2} c d^{2} e^{2} + 60 B b c^{2} d^{3} e - 35 B c^{3} d^{4}\right ) \log{\left (d + e x \right )}}{e^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.25406, size = 784, normalized size = 1.86 \begin{align*}{\left (35 \, B c^{3} d^{4} - 60 \, B b c^{2} d^{3} e - 20 \, A c^{3} d^{3} e + 30 \, B b^{2} c d^{2} e^{2} + 30 \, A b c^{2} d^{2} e^{2} - 4 \, B b^{3} d e^{3} - 12 \, A b^{2} c d e^{3} + A b^{3} e^{4}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{12} \,{\left (3 \, B c^{3} x^{4} e^{12} - 16 \, B c^{3} d x^{3} e^{11} + 60 \, B c^{3} d^{2} x^{2} e^{10} - 240 \, B c^{3} d^{3} x e^{9} + 12 \, B b c^{2} x^{3} e^{12} + 4 \, A c^{3} x^{3} e^{12} - 72 \, B b c^{2} d x^{2} e^{11} - 24 \, A c^{3} d x^{2} e^{11} + 360 \, B b c^{2} d^{2} x e^{10} + 120 \, A c^{3} d^{2} x e^{10} + 18 \, B b^{2} c x^{2} e^{12} + 18 \, A b c^{2} x^{2} e^{12} - 144 \, B b^{2} c d x e^{11} - 144 \, A b c^{2} d x e^{11} + 12 \, B b^{3} x e^{12} + 36 \, A b^{2} c x e^{12}\right )} e^{\left (-16\right )} + \frac{{\left (107 \, B c^{3} d^{7} - 222 \, B b c^{2} d^{6} e - 74 \, A c^{3} d^{6} e + 141 \, B b^{2} c d^{5} e^{2} + 141 \, A b c^{2} d^{5} e^{2} - 26 \, B b^{3} d^{4} e^{3} - 78 \, A b^{2} c d^{4} e^{3} + 11 \, A b^{3} d^{3} e^{4} + 18 \,{\left (7 \, B c^{3} d^{5} e^{2} - 15 \, B b c^{2} d^{4} e^{3} - 5 \, A c^{3} d^{4} e^{3} + 10 \, B b^{2} c d^{3} e^{4} + 10 \, A b c^{2} d^{3} e^{4} - 2 \, B b^{3} d^{2} e^{5} - 6 \, A b^{2} c d^{2} e^{5} + A b^{3} d e^{6}\right )} x^{2} + 3 \,{\left (77 \, B c^{3} d^{6} e - 162 \, B b c^{2} d^{5} e^{2} - 54 \, A c^{3} d^{5} e^{2} + 105 \, B b^{2} c d^{4} e^{3} + 105 \, A b c^{2} d^{4} e^{3} - 20 \, B b^{3} d^{3} e^{4} - 60 \, A b^{2} c d^{3} e^{4} + 9 \, A b^{3} d^{2} e^{5}\right )} x\right )} e^{\left (-8\right )}}{6 \,{\left (x e + d\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]