Optimal. Leaf size=308 \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (-3 a B e-A b e+4 b B d)}{11 e^5 (a+b x)}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e) (-a B e-A b e+2 b B d)}{3 e^5 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3 (B d-A e)}{5 e^5 (a+b x)}+\frac{2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^5 (a+b x)} \]
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Rubi [A] time = 0.143681, antiderivative size = 308, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {770, 77} \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (-3 a B e-A b e+4 b B d)}{11 e^5 (a+b x)}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e) (-a B e-A b e+2 b B d)}{3 e^5 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3 (B d-A e)}{5 e^5 (a+b x)}+\frac{2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^5 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 77
Rubi steps
\begin{align*} \int (A+B x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^3 (A+B x) (d+e x)^{3/2} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (-\frac{b^3 (b d-a e)^3 (-B d+A e) (d+e x)^{3/2}}{e^4}+\frac{b^3 (b d-a e)^2 (-4 b B d+3 A b e+a B e) (d+e x)^{5/2}}{e^4}-\frac{3 b^4 (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{7/2}}{e^4}+\frac{b^5 (-4 b B d+A b e+3 a B e) (d+e x)^{9/2}}{e^4}+\frac{b^6 B (d+e x)^{11/2}}{e^4}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{2 (b d-a e)^3 (B d-A e) (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x)}-\frac{2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}+\frac{2 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x)}-\frac{2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x)}+\frac{2 b^3 B (d+e x)^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}{13 e^5 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.211159, size = 163, normalized size = 0.53 \[ \frac{2 \left ((a+b x)^2\right )^{3/2} (d+e x)^{5/2} \left (-1365 b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)+5005 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)-2145 (d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)+3003 (b d-a e)^3 (B d-A e)+1155 b^3 B (d+e x)^4\right )}{15015 e^5 (a+b x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 317, normalized size = 1. \begin{align*}{\frac{2310\,B{x}^{4}{b}^{3}{e}^{4}+2730\,A{x}^{3}{b}^{3}{e}^{4}+8190\,B{x}^{3}a{b}^{2}{e}^{4}-1680\,B{x}^{3}{b}^{3}d{e}^{3}+10010\,A{x}^{2}a{b}^{2}{e}^{4}-1820\,A{x}^{2}{b}^{3}d{e}^{3}+10010\,B{x}^{2}{a}^{2}b{e}^{4}-5460\,B{x}^{2}a{b}^{2}d{e}^{3}+1120\,B{x}^{2}{b}^{3}{d}^{2}{e}^{2}+12870\,Ax{a}^{2}b{e}^{4}-5720\,Axa{b}^{2}d{e}^{3}+1040\,Ax{b}^{3}{d}^{2}{e}^{2}+4290\,Bx{a}^{3}{e}^{4}-5720\,Bx{a}^{2}bd{e}^{3}+3120\,Bxa{b}^{2}{d}^{2}{e}^{2}-640\,Bx{b}^{3}{d}^{3}e+6006\,A{a}^{3}{e}^{4}-5148\,Ad{e}^{3}{a}^{2}b+2288\,Aa{b}^{2}{d}^{2}{e}^{2}-416\,A{b}^{3}{d}^{3}e-1716\,Bd{e}^{3}{a}^{3}+2288\,B{a}^{2}b{d}^{2}{e}^{2}-1248\,Ba{b}^{2}{d}^{3}e+256\,B{b}^{3}{d}^{4}}{15015\,{e}^{5} \left ( bx+a \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.03378, size = 659, normalized size = 2.14 \begin{align*} \frac{2 \,{\left (105 \, b^{3} e^{5} x^{5} - 16 \, b^{3} d^{5} + 88 \, a b^{2} d^{4} e - 198 \, a^{2} b d^{3} e^{2} + 231 \, a^{3} d^{2} e^{3} + 35 \,{\left (4 \, b^{3} d e^{4} + 11 \, a b^{2} e^{5}\right )} x^{4} + 5 \,{\left (b^{3} d^{2} e^{3} + 110 \, a b^{2} d e^{4} + 99 \, a^{2} b e^{5}\right )} x^{3} - 3 \,{\left (2 \, b^{3} d^{3} e^{2} - 11 \, a b^{2} d^{2} e^{3} - 264 \, a^{2} b d e^{4} - 77 \, a^{3} e^{5}\right )} x^{2} +{\left (8 \, b^{3} d^{4} e - 44 \, a b^{2} d^{3} e^{2} + 99 \, a^{2} b d^{2} e^{3} + 462 \, a^{3} d e^{4}\right )} x\right )} \sqrt{e x + d} A}{1155 \, e^{4}} + \frac{2 \,{\left (1155 \, b^{3} e^{6} x^{6} + 128 \, b^{3} d^{6} - 624 \, a b^{2} d^{5} e + 1144 \, a^{2} b d^{4} e^{2} - 858 \, a^{3} d^{3} e^{3} + 105 \,{\left (14 \, b^{3} d e^{5} + 39 \, a b^{2} e^{6}\right )} x^{5} + 35 \,{\left (b^{3} d^{2} e^{4} + 156 \, a b^{2} d e^{5} + 143 \, a^{2} b e^{6}\right )} x^{4} - 5 \,{\left (8 \, b^{3} d^{3} e^{3} - 39 \, a b^{2} d^{2} e^{4} - 1430 \, a^{2} b d e^{5} - 429 \, a^{3} e^{6}\right )} x^{3} + 3 \,{\left (16 \, b^{3} d^{4} e^{2} - 78 \, a b^{2} d^{3} e^{3} + 143 \, a^{2} b d^{2} e^{4} + 1144 \, a^{3} d e^{5}\right )} x^{2} -{\left (64 \, b^{3} d^{5} e - 312 \, a b^{2} d^{4} e^{2} + 572 \, a^{2} b d^{3} e^{3} - 429 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt{e x + d} B}{15015 \, e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26392, size = 994, normalized size = 3.23 \begin{align*} \frac{2 \,{\left (1155 \, B b^{3} e^{6} x^{6} + 128 \, B b^{3} d^{6} + 3003 \, A a^{3} d^{2} e^{4} - 208 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} e + 1144 \,{\left (B a^{2} b + A a b^{2}\right )} d^{4} e^{2} - 858 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{3} + 105 \,{\left (14 \, B b^{3} d e^{5} + 13 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{6}\right )} x^{5} + 35 \,{\left (B b^{3} d^{2} e^{4} + 52 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{5} + 143 \,{\left (B a^{2} b + A a b^{2}\right )} e^{6}\right )} x^{4} - 5 \,{\left (8 \, B b^{3} d^{3} e^{3} - 13 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{4} - 1430 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{5} - 429 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{6}\right )} x^{3} + 3 \,{\left (16 \, B b^{3} d^{4} e^{2} + 1001 \, A a^{3} e^{6} - 26 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{3} + 143 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{4} + 1144 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{5}\right )} x^{2} -{\left (64 \, B b^{3} d^{5} e - 6006 \, A a^{3} d e^{5} - 104 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e^{2} + 572 \,{\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{3} - 429 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{4}\right )} x\right )} \sqrt{e x + d}}{15015 \, e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (A + B x\right ) \left (d + e x\right )^{\frac{3}{2}} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2452, size = 1220, normalized size = 3.96 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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