Optimal. Leaf size=263 \[ -\frac{2 (d+e x)^{3/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{3 e^6}+\frac{2 \sqrt{d+e x} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^6}+\frac{2 d^2 (B d-A e) (c d-b e)^2}{3 e^6 (d+e x)^{3/2}}-\frac{2 c (d+e x)^{5/2} (-A c e-2 b B e+5 B c d)}{5 e^6}-\frac{2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 \sqrt{d+e x}}+\frac{2 B c^2 (d+e x)^{7/2}}{7 e^6} \]
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Rubi [A] time = 0.15354, 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} \[ -\frac{2 (d+e x)^{3/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{3 e^6}+\frac{2 \sqrt{d+e x} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^6}+\frac{2 d^2 (B d-A e) (c d-b e)^2}{3 e^6 (d+e x)^{3/2}}-\frac{2 c (d+e x)^{5/2} (-A c e-2 b B e+5 B c d)}{5 e^6}-\frac{2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 \sqrt{d+e x}}+\frac{2 B c^2 (d+e x)^{7/2}}{7 e^6} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx &=\int \left (-\frac{d^2 (B d-A e) (c d-b e)^2}{e^5 (d+e x)^{5/2}}+\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^5 (d+e x)^{3/2}}+\frac{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 )}{e^5 \sqrt{d+e x}}+\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 ) \sqrt{d+e x}}{e^5}+\frac{c (-5 B c d+2 b B e+A c e) (d+e x)^{3/2}}{e^5}+\frac{B c^2 (d+e x)^{5/2}}{e^5}\right ) \, dx\\ &=\frac{2 d^2 (B d-A e) (c d-b e)^2}{3 e^6 (d+e x)^{3/2}}-\frac{2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 \sqrt{d+e x}}+\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 ) \sqrt{d+e x}}{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)^{3/2}}{3 e^6}-\frac{2 c (5 B c d-2 b B e-A c e) (d+e x)^{5/2}}{5 e^6}+\frac{2 B c^2 (d+e x)^{7/2}}{7 e^6}\\ \end{align*}
Mathematica [A] time = 0.16832, size = 271, normalized size = 1.03 \[ \frac{2 \left (7 A e \left (5 b^2 e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+10 b c e \left (-24 d^2 e x-16 d^3-6 d e^2 x^2+e^3 x^3\right )+c^2 \left (48 d^2 e^2 x^2+192 d^3 e x+128 d^4-8 d e^3 x^3+3 e^4 x^4\right )\right )+B \left (35 b^2 e^2 \left (-24 d^2 e x-16 d^3-6 d e^2 x^2+e^3 x^3\right )+14 b c e \left (48 d^2 e^2 x^2+192 d^3 e x+128 d^4-8 d e^3 x^3+3 e^4 x^4\right )-5 c^2 \left (96 d^3 e^2 x^2-16 d^2 e^3 x^3+384 d^4 e x+256 d^5+6 d e^4 x^4-3 e^5 x^5\right )\right )\right )}{105 e^6 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 341, normalized size = 1.3 \begin{align*}{\frac{30\,B{c}^{2}{x}^{5}{e}^{5}+42\,A{c}^{2}{e}^{5}{x}^{4}+84\,Bbc{e}^{5}{x}^{4}-60\,B{c}^{2}d{e}^{4}{x}^{4}+140\,Abc{e}^{5}{x}^{3}-112\,A{c}^{2}d{e}^{4}{x}^{3}+70\,B{b}^{2}{e}^{5}{x}^{3}-224\,Bbcd{e}^{4}{x}^{3}+160\,B{c}^{2}{d}^{2}{e}^{3}{x}^{3}+210\,A{b}^{2}{e}^{5}{x}^{2}-840\,Abcd{e}^{4}{x}^{2}+672\,A{c}^{2}{d}^{2}{e}^{3}{x}^{2}-420\,B{b}^{2}d{e}^{4}{x}^{2}+1344\,Bbc{d}^{2}{e}^{3}{x}^{2}-960\,B{c}^{2}{d}^{3}{e}^{2}{x}^{2}+840\,A{b}^{2}d{e}^{4}x-3360\,Abc{d}^{2}{e}^{3}x+2688\,A{c}^{2}{d}^{3}{e}^{2}x-1680\,B{b}^{2}{d}^{2}{e}^{3}x+5376\,Bbc{d}^{3}{e}^{2}x-3840\,B{c}^{2}{d}^{4}ex+560\,A{b}^{2}{d}^{2}{e}^{3}-2240\,Abc{d}^{3}{e}^{2}+1792\,A{c}^{2}{d}^{4}e-1120\,B{b}^{2}{d}^{3}{e}^{2}+3584\,Bbc{d}^{4}e-2560\,B{c}^{2}{d}^{5}}{105\,{e}^{6}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09155, size = 401, normalized size = 1.52 \begin{align*} \frac{2 \,{\left (\frac{15 \,{\left (e x + d\right )}^{\frac{7}{2}} B c^{2} - 21 \,{\left (5 \, B c^{2} d -{\left (2 \, B b c + A c^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\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{3}{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 )} \sqrt{e x + d}}{e^{5}} + \frac{35 \,{\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} - 3 \,{\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 )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{5}}\right )}}{105 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81972, size = 691, normalized size = 2.63 \begin{align*} \frac{2 \,{\left (15 \, B c^{2} e^{5} x^{5} - 1280 \, B c^{2} d^{5} + 280 \, A b^{2} d^{2} e^{3} + 896 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e - 560 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - 3 \,{\left (10 \, B c^{2} d e^{4} - 7 \,{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} +{\left (80 \, B c^{2} d^{2} e^{3} - 56 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} + 35 \,{\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} - 3 \,{\left (160 \, B c^{2} d^{3} e^{2} - 35 \, A b^{2} e^{5} - 112 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 70 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} - 12 \,{\left (160 \, B c^{2} d^{4} e - 35 \, A b^{2} d e^{4} - 112 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 70 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 70.2564, size = 292, normalized size = 1.11 \begin{align*} \frac{2 B c^{2} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{6}} + \frac{2 d^{2} \left (- A e + B d\right ) \left (b e - c d\right )^{2}}{3 e^{6} \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 d \left (b e - c d\right ) \left (- 2 A b e^{2} + 4 A c d e + 3 B b d e - 5 B c d^{2}\right )}{e^{6} \sqrt{d + e x}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (2 A c^{2} e + 4 B b c e - 10 B c^{2} d\right )}{5 e^{6}} + \frac{\left (d + e x\right )^{\frac{3}{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 )}{3 e^{6}} + \frac{\sqrt{d + e x} \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 )}{e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26866, size = 576, normalized size = 2.19 \begin{align*} \frac{2}{105} \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} B c^{2} e^{36} - 105 \,{\left (x e + d\right )}^{\frac{5}{2}} B c^{2} d e^{36} + 350 \,{\left (x e + d\right )}^{\frac{3}{2}} B c^{2} d^{2} e^{36} - 1050 \, \sqrt{x e + d} B c^{2} d^{3} e^{36} + 42 \,{\left (x e + d\right )}^{\frac{5}{2}} B b c e^{37} + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} A c^{2} e^{37} - 280 \,{\left (x e + d\right )}^{\frac{3}{2}} B b c d e^{37} - 140 \,{\left (x e + d\right )}^{\frac{3}{2}} A c^{2} d e^{37} + 1260 \, \sqrt{x e + d} B b c d^{2} e^{37} + 630 \, \sqrt{x e + d} A c^{2} d^{2} e^{37} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} e^{38} + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} A b c e^{38} - 315 \, \sqrt{x e + d} B b^{2} d e^{38} - 630 \, \sqrt{x e + d} A b c d e^{38} + 105 \, \sqrt{x e + d} A b^{2} e^{39}\right )} e^{\left (-42\right )} - \frac{2 \,{\left (15 \,{\left (x e + d\right )} B c^{2} d^{4} - B c^{2} d^{5} - 24 \,{\left (x e + d\right )} B b c d^{3} e - 12 \,{\left (x e + d\right )} A c^{2} d^{3} e + 2 \, B b c d^{4} e + A c^{2} d^{4} e + 9 \,{\left (x e + d\right )} B b^{2} d^{2} e^{2} + 18 \,{\left (x e + d\right )} A b c d^{2} e^{2} - B b^{2} d^{3} e^{2} - 2 \, A b c d^{3} e^{2} - 6 \,{\left (x e + d\right )} A b^{2} d e^{3} + A b^{2} d^{2} e^{3}\right )} e^{\left (-6\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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