Optimal. Leaf size=246 \[ \frac{5 (b d-a e)^3 (a B e-8 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{64 b^{3/2} e^{9/2}}-\frac{(a+b x)^{5/2} \sqrt{d+e x} (a B e-8 A b e+7 b B d)}{24 b e^2}+\frac{5 (a+b x)^{3/2} \sqrt{d+e x} (b d-a e) (a B e-8 A b e+7 b B d)}{96 b e^3}-\frac{5 \sqrt{a+b x} \sqrt{d+e x} (b d-a e)^2 (a B e-8 A b e+7 b B d)}{64 b e^4}+\frac{B (a+b x)^{7/2} \sqrt{d+e x}}{4 b e} \]
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Rubi [A] time = 0.204348, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{5 (b d-a e)^3 (a B e-8 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{64 b^{3/2} e^{9/2}}-\frac{(a+b x)^{5/2} \sqrt{d+e x} (a B e-8 A b e+7 b B d)}{24 b e^2}+\frac{5 (a+b x)^{3/2} \sqrt{d+e x} (b d-a e) (a B e-8 A b e+7 b B d)}{96 b e^3}-\frac{5 \sqrt{a+b x} \sqrt{d+e x} (b d-a e)^2 (a B e-8 A b e+7 b B d)}{64 b e^4}+\frac{B (a+b x)^{7/2} \sqrt{d+e x}}{4 b e} \]
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)^{5/2} (A+B x)}{\sqrt{d+e x}} \, dx &=\frac{B (a+b x)^{7/2} \sqrt{d+e x}}{4 b e}+\frac{\left (4 A b e-B \left (\frac{7 b d}{2}+\frac{a e}{2}\right )\right ) \int \frac{(a+b x)^{5/2}}{\sqrt{d+e x}} \, dx}{4 b e}\\ &=-\frac{(7 b B d-8 A b e+a B e) (a+b x)^{5/2} \sqrt{d+e x}}{24 b e^2}+\frac{B (a+b x)^{7/2} \sqrt{d+e x}}{4 b e}+\frac{(5 (b d-a e) (7 b B d-8 A b e+a B e)) \int \frac{(a+b x)^{3/2}}{\sqrt{d+e x}} \, dx}{48 b e^2}\\ &=\frac{5 (b d-a e) (7 b B d-8 A b e+a B e) (a+b x)^{3/2} \sqrt{d+e x}}{96 b e^3}-\frac{(7 b B d-8 A b e+a B e) (a+b x)^{5/2} \sqrt{d+e x}}{24 b e^2}+\frac{B (a+b x)^{7/2} \sqrt{d+e x}}{4 b e}-\frac{\left (5 (b d-a e)^2 (7 b B d-8 A b e+a B e)\right ) \int \frac{\sqrt{a+b x}}{\sqrt{d+e x}} \, dx}{64 b e^3}\\ &=-\frac{5 (b d-a e)^2 (7 b B d-8 A b e+a B e) \sqrt{a+b x} \sqrt{d+e x}}{64 b e^4}+\frac{5 (b d-a e) (7 b B d-8 A b e+a B e) (a+b x)^{3/2} \sqrt{d+e x}}{96 b e^3}-\frac{(7 b B d-8 A b e+a B e) (a+b x)^{5/2} \sqrt{d+e x}}{24 b e^2}+\frac{B (a+b x)^{7/2} \sqrt{d+e x}}{4 b e}+\frac{\left (5 (b d-a e)^3 (7 b B d-8 A b e+a B e)\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{d+e x}} \, dx}{128 b e^4}\\ &=-\frac{5 (b d-a e)^2 (7 b B d-8 A b e+a B e) \sqrt{a+b x} \sqrt{d+e x}}{64 b e^4}+\frac{5 (b d-a e) (7 b B d-8 A b e+a B e) (a+b x)^{3/2} \sqrt{d+e x}}{96 b e^3}-\frac{(7 b B d-8 A b e+a B e) (a+b x)^{5/2} \sqrt{d+e x}}{24 b e^2}+\frac{B (a+b x)^{7/2} \sqrt{d+e x}}{4 b e}+\frac{\left (5 (b d-a e)^3 (7 b B d-8 A b e+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 )}{64 b^2 e^4}\\ &=-\frac{5 (b d-a e)^2 (7 b B d-8 A b e+a B e) \sqrt{a+b x} \sqrt{d+e x}}{64 b e^4}+\frac{5 (b d-a e) (7 b B d-8 A b e+a B e) (a+b x)^{3/2} \sqrt{d+e x}}{96 b e^3}-\frac{(7 b B d-8 A b e+a B e) (a+b x)^{5/2} \sqrt{d+e x}}{24 b e^2}+\frac{B (a+b x)^{7/2} \sqrt{d+e x}}{4 b e}+\frac{\left (5 (b d-a e)^3 (7 b B d-8 A b e+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 )}{64 b^2 e^4}\\ &=-\frac{5 (b d-a e)^2 (7 b B d-8 A b e+a B e) \sqrt{a+b x} \sqrt{d+e x}}{64 b e^4}+\frac{5 (b d-a e) (7 b B d-8 A b e+a B e) (a+b x)^{3/2} \sqrt{d+e x}}{96 b e^3}-\frac{(7 b B d-8 A b e+a B e) (a+b x)^{5/2} \sqrt{d+e x}}{24 b e^2}+\frac{B (a+b x)^{7/2} \sqrt{d+e x}}{4 b e}+\frac{5 (b d-a e)^3 (7 b B d-8 A b e+a B e) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{64 b^{3/2} e^{9/2}}\\ \end{align*}
Mathematica [A] time = 1.25706, size = 297, normalized size = 1.21 \[ \frac{\sqrt{d+e x} \left (\frac{b (-a B e+8 A b e-7 b B d) \left (8 b^3 e^3 (a+b x)^3 \sqrt{b d-a e} \sqrt{\frac{b (d+e x)}{b d-a e}}-10 b^3 e^2 (a+b x)^2 (b d-a e)^{3/2} \sqrt{\frac{b (d+e x)}{b d-a e}}+15 b^3 e (a+b x) (b d-a e)^{5/2} \sqrt{\frac{b (d+e x)}{b d-a e}}-15 b^3 \sqrt{e} \sqrt{a+b x} (b d-a e)^3 \sinh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b d-a e}}\right )\right )}{3 \sqrt{b d-a e} \sqrt{\frac{b (d+e x)}{b d-a e}}}+16 b^4 B e^4 (a+b x)^4\right )}{64 b^5 e^5 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.021, size = 968, normalized size = 3.9 \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 = 2.01936, size = 1716, normalized size = 6.98 \begin{align*} \text{result too large to display} \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 [A] time = 1.83586, size = 527, normalized size = 2.14 \begin{align*} \frac{{\left (\sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{6 \,{\left (b x + a\right )} B e^{\left (-1\right )}}{b^{2}} - \frac{{\left (7 \, B b^{3} d e^{5} + B a b^{2} e^{6} - 8 \, A b^{3} e^{6}\right )} e^{\left (-7\right )}}{b^{4}}\right )} + \frac{5 \,{\left (7 \, B b^{4} d^{2} e^{4} - 6 \, B a b^{3} d e^{5} - 8 \, A b^{4} d e^{5} - B a^{2} b^{2} e^{6} + 8 \, A a b^{3} e^{6}\right )} e^{\left (-7\right )}}{b^{4}}\right )} - \frac{15 \,{\left (7 \, B b^{5} d^{3} e^{3} - 13 \, B a b^{4} d^{2} e^{4} - 8 \, A b^{5} d^{2} e^{4} + 5 \, B a^{2} b^{3} d e^{5} + 16 \, A a b^{4} d e^{5} + B a^{3} b^{2} e^{6} - 8 \, A a^{2} b^{3} e^{6}\right )} e^{\left (-7\right )}}{b^{4}}\right )} \sqrt{b x + a} - \frac{15 \,{\left (7 \, B b^{4} d^{4} - 20 \, B a b^{3} d^{3} e - 8 \, A b^{4} d^{3} e + 18 \, B a^{2} b^{2} d^{2} e^{2} + 24 \, A a b^{3} d^{2} e^{2} - 4 \, B a^{3} b d e^{3} - 24 \, A a^{2} b^{2} d e^{3} - B a^{4} e^{4} + 8 \, A a^{3} b e^{4}\right )} e^{\left (-\frac{9}{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 )}{b^{\frac{3}{2}}}\right )} b}{192 \,{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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