Optimal. Leaf size=263 \[ -\frac{2 (d+e x)^{5/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{5 e^6}+\frac{2 (d+e x)^{3/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{3 e^6}+\frac{2 d^2 (B d-A e) (c d-b e)^2}{e^6 \sqrt{d+e x}}-\frac{2 c (d+e x)^{7/2} (-A c e-2 b B e+5 B c d)}{7 e^6}+\frac{2 d \sqrt{d+e x} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6}+\frac{2 B c^2 (d+e x)^{9/2}}{9 e^6} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.159456, antiderivative size = 263, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {771} \[ -\frac{2 (d+e x)^{5/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{5 e^6}+\frac{2 (d+e x)^{3/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{3 e^6}+\frac{2 d^2 (B d-A e) (c d-b e)^2}{e^6 \sqrt{d+e x}}-\frac{2 c (d+e x)^{7/2} (-A c e-2 b B e+5 B c d)}{7 e^6}+\frac{2 d \sqrt{d+e x} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6}+\frac{2 B c^2 (d+e x)^{9/2}}{9 e^6} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 771
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx &=\int \left (-\frac{d^2 (B d-A e) (c d-b e)^2}{e^5 (d+e x)^{3/2}}+\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^5 \sqrt{d+e x}}+\frac{\left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) \sqrt{d+e x}}{e^5}+\frac{\left (-2 A c e (2 c d-b e)+B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{3/2}}{e^5}+\frac{c (-5 B c d+2 b B e+A c e) (d+e x)^{5/2}}{e^5}+\frac{B c^2 (d+e x)^{7/2}}{e^5}\right ) \, dx\\ &=\frac{2 d^2 (B d-A e) (c d-b e)^2}{e^6 \sqrt{d+e x}}+\frac{2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) \sqrt{d+e x}}{e^6}+\frac{2 \left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{3/2}}{3 e^6}-\frac{2 \left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{5/2}}{5 e^6}-\frac{2 c (5 B c d-2 b B e-A c e) (d+e x)^{7/2}}{7 e^6}+\frac{2 B c^2 (d+e x)^{9/2}}{9 e^6}\\ \end{align*}
Mathematica [A] time = 0.176328, size = 273, normalized size = 1.04 \[ \frac{2 B \left (63 b^2 e^2 \left (8 d^2 e x+16 d^3-2 d e^2 x^2+e^3 x^3\right )+18 b c e \left (16 d^2 e^2 x^2-64 d^3 e x-128 d^4-8 d e^3 x^3+5 e^4 x^4\right )+5 c^2 \left (-32 d^3 e^2 x^2+16 d^2 e^3 x^3+128 d^4 e x+256 d^5-10 d e^4 x^4+7 e^5 x^5\right )\right )-6 A e \left (35 b^2 e^2 \left (8 d^2+4 d e x-e^2 x^2\right )-42 b c e \left (8 d^2 e x+16 d^3-2 d e^2 x^2+e^3 x^3\right )+3 c^2 \left (-16 d^2 e^2 x^2+64 d^3 e x+128 d^4+8 d e^3 x^3-5 e^4 x^4\right )\right )}{315 e^6 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 341, normalized size = 1.3 \begin{align*} -{\frac{-70\,B{c}^{2}{x}^{5}{e}^{5}-90\,A{c}^{2}{e}^{5}{x}^{4}-180\,Bbc{e}^{5}{x}^{4}+100\,B{c}^{2}d{e}^{4}{x}^{4}-252\,Abc{e}^{5}{x}^{3}+144\,A{c}^{2}d{e}^{4}{x}^{3}-126\,B{b}^{2}{e}^{5}{x}^{3}+288\,Bbcd{e}^{4}{x}^{3}-160\,B{c}^{2}{d}^{2}{e}^{3}{x}^{3}-210\,A{b}^{2}{e}^{5}{x}^{2}+504\,Abcd{e}^{4}{x}^{2}-288\,A{c}^{2}{d}^{2}{e}^{3}{x}^{2}+252\,B{b}^{2}d{e}^{4}{x}^{2}-576\,Bbc{d}^{2}{e}^{3}{x}^{2}+320\,B{c}^{2}{d}^{3}{e}^{2}{x}^{2}+840\,A{b}^{2}d{e}^{4}x-2016\,Abc{d}^{2}{e}^{3}x+1152\,A{c}^{2}{d}^{3}{e}^{2}x-1008\,B{b}^{2}{d}^{2}{e}^{3}x+2304\,Bbc{d}^{3}{e}^{2}x-1280\,B{c}^{2}{d}^{4}ex+1680\,A{b}^{2}{d}^{2}{e}^{3}-4032\,Abc{d}^{3}{e}^{2}+2304\,A{c}^{2}{d}^{4}e-2016\,B{b}^{2}{d}^{3}{e}^{2}+4608\,Bbc{d}^{4}e-2560\,B{c}^{2}{d}^{5}}{315\,{e}^{6}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.11428, size = 404, normalized size = 1.54 \begin{align*} \frac{2 \,{\left (\frac{35 \,{\left (e x + d\right )}^{\frac{9}{2}} B c^{2} - 45 \,{\left (5 \, B c^{2} d -{\left (2 \, B b c + A c^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 63 \,{\left (10 \, B c^{2} d^{2} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d e +{\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 105 \,{\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 315 \,{\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )} \sqrt{e x + d}}{e^{5}} + \frac{315 \,{\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} -{\left (2 \, B b c + A c^{2}\right )} d^{4} e +{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}\right )}}{\sqrt{e x + d} e^{5}}\right )}}{315 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.75769, size = 674, normalized size = 2.56 \begin{align*} \frac{2 \,{\left (35 \, B c^{2} e^{5} x^{5} + 1280 \, B c^{2} d^{5} - 840 \, A b^{2} d^{2} e^{3} - 1152 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 1008 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - 5 \,{\left (10 \, B c^{2} d e^{4} - 9 \,{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} +{\left (80 \, B c^{2} d^{2} e^{3} - 72 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} + 63 \,{\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} -{\left (160 \, B c^{2} d^{3} e^{2} - 105 \, A b^{2} e^{5} - 144 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 126 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 4 \,{\left (160 \, B c^{2} d^{4} e - 105 \, A b^{2} d e^{4} - 144 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 126 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{315 \,{\left (e^{7} x + d e^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 69.807, size = 321, normalized size = 1.22 \begin{align*} \frac{2 B c^{2} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{6}} + \frac{2 d^{2} \left (- A e + B d\right ) \left (b e - c d\right )^{2}}{e^{6} \sqrt{d + e x}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (2 A c^{2} e + 4 B b c e - 10 B c^{2} d\right )}{7 e^{6}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (4 A b c e^{2} - 8 A c^{2} d e + 2 B b^{2} e^{2} - 16 B b c d e + 20 B c^{2} d^{2}\right )}{5 e^{6}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (2 A b^{2} e^{3} - 12 A b c d e^{2} + 12 A c^{2} d^{2} e - 6 B b^{2} d e^{2} + 24 B b c d^{2} e - 20 B c^{2} d^{3}\right )}{3 e^{6}} + \frac{\sqrt{d + e x} \left (- 4 A b^{2} d e^{3} + 12 A b c d^{2} e^{2} - 8 A c^{2} d^{3} e + 6 B b^{2} d^{2} e^{2} - 16 B b c d^{3} e + 10 B c^{2} d^{4}\right )}{e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.24678, size = 595, normalized size = 2.26 \begin{align*} \frac{2}{315} \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} B c^{2} e^{48} - 225 \,{\left (x e + d\right )}^{\frac{7}{2}} B c^{2} d e^{48} + 630 \,{\left (x e + d\right )}^{\frac{5}{2}} B c^{2} d^{2} e^{48} - 1050 \,{\left (x e + d\right )}^{\frac{3}{2}} B c^{2} d^{3} e^{48} + 1575 \, \sqrt{x e + d} B c^{2} d^{4} e^{48} + 90 \,{\left (x e + d\right )}^{\frac{7}{2}} B b c e^{49} + 45 \,{\left (x e + d\right )}^{\frac{7}{2}} A c^{2} e^{49} - 504 \,{\left (x e + d\right )}^{\frac{5}{2}} B b c d e^{49} - 252 \,{\left (x e + d\right )}^{\frac{5}{2}} A c^{2} d e^{49} + 1260 \,{\left (x e + d\right )}^{\frac{3}{2}} B b c d^{2} e^{49} + 630 \,{\left (x e + d\right )}^{\frac{3}{2}} A c^{2} d^{2} e^{49} - 2520 \, \sqrt{x e + d} B b c d^{3} e^{49} - 1260 \, \sqrt{x e + d} A c^{2} d^{3} e^{49} + 63 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{2} e^{50} + 126 \,{\left (x e + d\right )}^{\frac{5}{2}} A b c e^{50} - 315 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} d e^{50} - 630 \,{\left (x e + d\right )}^{\frac{3}{2}} A b c d e^{50} + 945 \, \sqrt{x e + d} B b^{2} d^{2} e^{50} + 1890 \, \sqrt{x e + d} A b c d^{2} e^{50} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{2} e^{51} - 630 \, \sqrt{x e + d} A b^{2} d e^{51}\right )} e^{\left (-54\right )} + \frac{2 \,{\left (B c^{2} d^{5} - 2 \, B b c d^{4} e - A c^{2} d^{4} e + B b^{2} d^{3} e^{2} + 2 \, A b c d^{3} e^{2} - A b^{2} d^{2} e^{3}\right )} e^{\left (-6\right )}}{\sqrt{x e + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]