Optimal. Leaf size=202 \[ -\frac{2 (a+b x)^{3/2} (-3 a B e-2 A b e+5 b B d)}{3 e^2 \sqrt{d+e x} (b d-a e)}+\frac{b \sqrt{a+b x} \sqrt{d+e x} (-3 a B e-2 A b e+5 b B d)}{e^3 (b d-a e)}-\frac{\sqrt{b} (-3 a B e-2 A b e+5 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{e^{7/2}}-\frac{2 (a+b x)^{5/2} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)} \]
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Rubi [A] time = 0.161261, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {78, 47, 50, 63, 217, 206} \[ -\frac{2 (a+b x)^{3/2} (-3 a B e-2 A b e+5 b B d)}{3 e^2 \sqrt{d+e x} (b d-a e)}+\frac{b \sqrt{a+b x} \sqrt{d+e x} (-3 a B e-2 A b e+5 b B d)}{e^3 (b d-a e)}-\frac{\sqrt{b} (-3 a B e-2 A b e+5 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{e^{7/2}}-\frac{2 (a+b x)^{5/2} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 78
Rule 47
Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2} (A+B x)}{(d+e x)^{5/2}} \, dx &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{3 e (b d-a e) (d+e x)^{3/2}}+\frac{(5 b B d-2 A b e-3 a B e) \int \frac{(a+b x)^{3/2}}{(d+e x)^{3/2}} \, dx}{3 e (b d-a e)}\\ &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{3 e (b d-a e) (d+e x)^{3/2}}-\frac{2 (5 b B d-2 A b e-3 a B e) (a+b x)^{3/2}}{3 e^2 (b d-a e) \sqrt{d+e x}}+\frac{(b (5 b B d-2 A b e-3 a B e)) \int \frac{\sqrt{a+b x}}{\sqrt{d+e x}} \, dx}{e^2 (b d-a e)}\\ &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{3 e (b d-a e) (d+e x)^{3/2}}-\frac{2 (5 b B d-2 A b e-3 a B e) (a+b x)^{3/2}}{3 e^2 (b d-a e) \sqrt{d+e x}}+\frac{b (5 b B d-2 A b e-3 a B e) \sqrt{a+b x} \sqrt{d+e x}}{e^3 (b d-a e)}-\frac{(b (5 b B d-2 A b e-3 a B e)) \int \frac{1}{\sqrt{a+b x} \sqrt{d+e x}} \, dx}{2 e^3}\\ &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{3 e (b d-a e) (d+e x)^{3/2}}-\frac{2 (5 b B d-2 A b e-3 a B e) (a+b x)^{3/2}}{3 e^2 (b d-a e) \sqrt{d+e x}}+\frac{b (5 b B d-2 A b e-3 a B e) \sqrt{a+b x} \sqrt{d+e x}}{e^3 (b d-a e)}-\frac{(5 b B d-2 A b e-3 a B e) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d-\frac{a e}{b}+\frac{e x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{e^3}\\ &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{3 e (b d-a e) (d+e x)^{3/2}}-\frac{2 (5 b B d-2 A b e-3 a B e) (a+b x)^{3/2}}{3 e^2 (b d-a e) \sqrt{d+e x}}+\frac{b (5 b B d-2 A b e-3 a B e) \sqrt{a+b x} \sqrt{d+e x}}{e^3 (b d-a e)}-\frac{(5 b B d-2 A b e-3 a B e) \operatorname{Subst}\left (\int \frac{1}{1-\frac{e x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{d+e x}}\right )}{e^3}\\ &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{3 e (b d-a e) (d+e x)^{3/2}}-\frac{2 (5 b B d-2 A b e-3 a B e) (a+b x)^{3/2}}{3 e^2 (b d-a e) \sqrt{d+e x}}+\frac{b (5 b B d-2 A b e-3 a B e) \sqrt{a+b x} \sqrt{d+e x}}{e^3 (b d-a e)}-\frac{\sqrt{b} (5 b B d-2 A b e-3 a B e) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{e^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.132795, size = 113, normalized size = 0.56 \[ \frac{2 (a+b x)^{5/2} \left (-\frac{\left (\frac{b (d+e x)}{b d-a e}\right )^{3/2} (-3 a B e-2 A b e+5 b B d) \, _2F_1\left (\frac{3}{2},\frac{5}{2};\frac{7}{2};\frac{e (a+b x)}{a e-b d}\right )}{b}-5 A e+5 B d\right )}{15 e (d+e x)^{3/2} (a e-b d)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.022, size = 698, normalized size = 3.5 \begin{align*} \text{result too large to display} \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 = 8.95025, size = 1200, normalized size = 5.94 \begin{align*} \left [-\frac{3 \,{\left (5 \, B b d^{3} -{\left (3 \, B a + 2 \, A b\right )} d^{2} e +{\left (5 \, B b d e^{2} -{\left (3 \, B a + 2 \, A b\right )} e^{3}\right )} x^{2} + 2 \,{\left (5 \, B b d^{2} e -{\left (3 \, B a + 2 \, A b\right )} d e^{2}\right )} x\right )} \sqrt{\frac{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^{2} x + b d e + a e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d} \sqrt{\frac{b}{e}} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right ) - 4 \,{\left (3 \, B b e^{2} x^{2} + 15 \, B b d^{2} - 2 \, A a e^{2} - 2 \,{\left (2 \, B a + 3 \, A b\right )} d e + 2 \,{\left (10 \, B b d e -{\left (3 \, B a + 4 \, A b\right )} e^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{12 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}}, \frac{3 \,{\left (5 \, B b d^{3} -{\left (3 \, B a + 2 \, A b\right )} d^{2} e +{\left (5 \, B b d e^{2} -{\left (3 \, B a + 2 \, A b\right )} e^{3}\right )} x^{2} + 2 \,{\left (5 \, B b d^{2} e -{\left (3 \, B a + 2 \, A b\right )} d e^{2}\right )} x\right )} \sqrt{-\frac{b}{e}} \arctan \left (\frac{{\left (2 \, b e x + b d + a e\right )} \sqrt{b x + a} \sqrt{e x + d} \sqrt{-\frac{b}{e}}}{2 \,{\left (b^{2} e x^{2} + a b d +{\left (b^{2} d + a b e\right )} x\right )}}\right ) + 2 \,{\left (3 \, B b e^{2} x^{2} + 15 \, B b d^{2} - 2 \, A a e^{2} - 2 \,{\left (2 \, B a + 3 \, A b\right )} d e + 2 \,{\left (10 \, B b d e -{\left (3 \, B a + 4 \, A b\right )} e^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{6 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.66163, size = 475, normalized size = 2.35 \begin{align*} \frac{{\left (5 \, B b d{\left | b \right |} - 3 \, B a{\left | b \right |} e - 2 \, A b{\left | b \right |} e\right )} e^{\left (-\frac{7}{2}\right )} \log \left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt{b}} + \frac{{\left ({\left (b x + a\right )}{\left (\frac{3 \,{\left (B b^{5} d{\left | b \right |} e^{4} - B a b^{4}{\left | b \right |} e^{5}\right )}{\left (b x + a\right )}}{b^{4} d e^{5} - a b^{3} e^{6}} + \frac{4 \,{\left (5 \, B b^{6} d^{2}{\left | b \right |} e^{3} - 8 \, B a b^{5} d{\left | b \right |} e^{4} - 2 \, A b^{6} d{\left | b \right |} e^{4} + 3 \, B a^{2} b^{4}{\left | b \right |} e^{5} + 2 \, A a b^{5}{\left | b \right |} e^{5}\right )}}{b^{4} d e^{5} - a b^{3} e^{6}}\right )} + \frac{3 \,{\left (5 \, B b^{7} d^{3}{\left | b \right |} e^{2} - 13 \, B a b^{6} d^{2}{\left | b \right |} e^{3} - 2 \, A b^{7} d^{2}{\left | b \right |} e^{3} + 11 \, B a^{2} b^{5} d{\left | b \right |} e^{4} + 4 \, A a b^{6} d{\left | b \right |} e^{4} - 3 \, B a^{3} b^{4}{\left | b \right |} e^{5} - 2 \, A a^{2} b^{5}{\left | b \right |} e^{5}\right )}}{b^{4} d e^{5} - a b^{3} e^{6}}\right )} \sqrt{b x + a}}{3 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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