Optimal. Leaf size=193 \[ -\frac{(b d-a e)^2 (5 a B e-6 A b e+b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{8 b^{7/2} e^{3/2}}-\frac{\sqrt{a+b x} (d+e x)^{3/2} (5 a B e-6 A b e+b B d)}{12 b^2 e}-\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e) (5 a B e-6 A b e+b B d)}{8 b^3 e}+\frac{B \sqrt{a+b x} (d+e x)^{5/2}}{3 b e} \]
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Rubi [A] time = 0.148768, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {80, 50, 63, 217, 206} \[ -\frac{(b d-a e)^2 (5 a B e-6 A b e+b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{8 b^{7/2} e^{3/2}}-\frac{\sqrt{a+b x} (d+e x)^{3/2} (5 a B e-6 A b e+b B d)}{12 b^2 e}-\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e) (5 a B e-6 A b e+b B d)}{8 b^3 e}+\frac{B \sqrt{a+b x} (d+e x)^{5/2}}{3 b e} \]
Antiderivative was successfully verified.
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Rule 80
Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)^{3/2}}{\sqrt{a+b x}} \, dx &=\frac{B \sqrt{a+b x} (d+e x)^{5/2}}{3 b e}+\frac{\left (3 A b e-B \left (\frac{b d}{2}+\frac{5 a e}{2}\right )\right ) \int \frac{(d+e x)^{3/2}}{\sqrt{a+b x}} \, dx}{3 b e}\\ &=-\frac{(b B d-6 A b e+5 a B e) \sqrt{a+b x} (d+e x)^{3/2}}{12 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{5/2}}{3 b e}-\frac{((b d-a e) (b B d-6 A b e+5 a B e)) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x}} \, dx}{8 b^2 e}\\ &=-\frac{(b d-a e) (b B d-6 A b e+5 a B e) \sqrt{a+b x} \sqrt{d+e x}}{8 b^3 e}-\frac{(b B d-6 A b e+5 a B e) \sqrt{a+b x} (d+e x)^{3/2}}{12 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{5/2}}{3 b e}-\frac{\left ((b d-a e)^2 (b B d-6 A b e+5 a B e)\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{d+e x}} \, dx}{16 b^3 e}\\ &=-\frac{(b d-a e) (b B d-6 A b e+5 a B e) \sqrt{a+b x} \sqrt{d+e x}}{8 b^3 e}-\frac{(b B d-6 A b e+5 a B e) \sqrt{a+b x} (d+e x)^{3/2}}{12 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{5/2}}{3 b e}-\frac{\left ((b d-a e)^2 (b B d-6 A b e+5 a B e)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d-\frac{a e}{b}+\frac{e x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{8 b^4 e}\\ &=-\frac{(b d-a e) (b B d-6 A b e+5 a B e) \sqrt{a+b x} \sqrt{d+e x}}{8 b^3 e}-\frac{(b B d-6 A b e+5 a B e) \sqrt{a+b x} (d+e x)^{3/2}}{12 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{5/2}}{3 b e}-\frac{\left ((b d-a e)^2 (b B d-6 A b e+5 a B e)\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{e x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{d+e x}}\right )}{8 b^4 e}\\ &=-\frac{(b d-a e) (b B d-6 A b e+5 a B e) \sqrt{a+b x} \sqrt{d+e x}}{8 b^3 e}-\frac{(b B d-6 A b e+5 a B e) \sqrt{a+b x} (d+e x)^{3/2}}{12 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{5/2}}{3 b e}-\frac{(b d-a e)^2 (b B d-6 A b e+5 a B e) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{8 b^{7/2} e^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.572491, size = 182, normalized size = 0.94 \[ \frac{\sqrt{d+e x} \left (\sqrt{e} \sqrt{a+b x} \left (15 a^2 B e^2-2 a b e (9 A e+11 B d+5 B e x)+b^2 \left (6 A e (5 d+2 e x)+B \left (3 d^2+14 d e x+8 e^2 x^2\right )\right )\right )-\frac{3 (b d-a e)^{3/2} (5 a B e-6 A b e+b B d) \sinh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b d-a e}}\right )}{\sqrt{\frac{b (d+e x)}{b d-a e}}}\right )}{24 b^3 e^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.019, size = 636, normalized size = 3.3 \begin{align*}{\frac{1}{48\,{b}^{3}e}\sqrt{ex+d}\sqrt{bx+a} \left ( 16\,B{x}^{2}{b}^{2}{e}^{2}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+18\,A\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{2}b{e}^{3}-36\,A\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) a{b}^{2}d{e}^{2}+18\,A\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){b}^{3}{d}^{2}e+24\,A\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}x{b}^{2}{e}^{2}-15\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{3}{e}^{3}+27\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{2}bd{e}^{2}-9\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) a{b}^{2}{d}^{2}e-3\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){b}^{3}{d}^{3}-20\,B\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}xab{e}^{2}+28\,B\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}x{b}^{2}de-36\,A\sqrt{be}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }ab{e}^{2}+60\,A\sqrt{be}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }{b}^{2}de+30\,B\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}{a}^{2}{e}^{2}-44\,B\sqrt{be}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }abde+6\,B\sqrt{be}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }{b}^{2}{d}^{2} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }}}{\frac{1}{\sqrt{be}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53768, size = 1215, normalized size = 6.3 \begin{align*} \left [-\frac{3 \,{\left (B b^{3} d^{3} + 3 \,{\left (B a b^{2} - 2 \, A b^{3}\right )} d^{2} e - 3 \,{\left (3 \, B a^{2} b - 4 \, A a b^{2}\right )} d e^{2} +{\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} e^{3}\right )} \sqrt{b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \,{\left (2 \, b e x + b d + a e\right )} \sqrt{b e} \sqrt{b x + a} \sqrt{e x + d} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right ) - 4 \,{\left (8 \, B b^{3} e^{3} x^{2} + 3 \, B b^{3} d^{2} e - 2 \,{\left (11 \, B a b^{2} - 15 \, A b^{3}\right )} d e^{2} + 3 \,{\left (5 \, B a^{2} b - 6 \, A a b^{2}\right )} e^{3} + 2 \,{\left (7 \, B b^{3} d e^{2} -{\left (5 \, B a b^{2} - 6 \, A b^{3}\right )} e^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{96 \, b^{4} e^{2}}, \frac{3 \,{\left (B b^{3} d^{3} + 3 \,{\left (B a b^{2} - 2 \, A b^{3}\right )} d^{2} e - 3 \,{\left (3 \, B a^{2} b - 4 \, A a b^{2}\right )} d e^{2} +{\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} e^{3}\right )} \sqrt{-b e} \arctan \left (\frac{{\left (2 \, b e x + b d + a e\right )} \sqrt{-b e} \sqrt{b x + a} \sqrt{e x + d}}{2 \,{\left (b^{2} e^{2} x^{2} + a b d e +{\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) + 2 \,{\left (8 \, B b^{3} e^{3} x^{2} + 3 \, B b^{3} d^{2} e - 2 \,{\left (11 \, B a b^{2} - 15 \, A b^{3}\right )} d e^{2} + 3 \,{\left (5 \, B a^{2} b - 6 \, A a b^{2}\right )} e^{3} + 2 \,{\left (7 \, B b^{3} d e^{2} -{\left (5 \, B a b^{2} - 6 \, A b^{3}\right )} e^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{48 \, b^{4} e^{2}}\right ] \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 \frac{\left (A + B x\right ) \left (d + e x\right )^{\frac{3}{2}}}{\sqrt{a + b x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.77904, size = 790, normalized size = 4.09 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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