Optimal. Leaf size=95 \[ \frac{2 (a+b x)^{3/2} (-5 a B e+2 A b e+3 b B d)}{15 e (d+e x)^{3/2} (b d-a e)^2}-\frac{2 (a+b x)^{3/2} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)} \]
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Rubi [A] time = 0.0509111, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {78, 37} \[ \frac{2 (a+b x)^{3/2} (-5 a B e+2 A b e+3 b B d)}{15 e (d+e x)^{3/2} (b d-a e)^2}-\frac{2 (a+b x)^{3/2} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 78
Rule 37
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x} (A+B x)}{(d+e x)^{7/2}} \, dx &=-\frac{2 (B d-A e) (a+b x)^{3/2}}{5 e (b d-a e) (d+e x)^{5/2}}+\frac{(3 b B d+2 A b e-5 a B e) \int \frac{\sqrt{a+b x}}{(d+e x)^{5/2}} \, dx}{5 e (b d-a e)}\\ &=-\frac{2 (B d-A e) (a+b x)^{3/2}}{5 e (b d-a e) (d+e x)^{5/2}}+\frac{2 (3 b B d+2 A b e-5 a B e) (a+b x)^{3/2}}{15 e (b d-a e)^2 (d+e x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.043971, size = 66, normalized size = 0.69 \[ \frac{2 (a+b x)^{3/2} (A (-3 a e+5 b d+2 b e x)+B (-2 a d-5 a e x+3 b d x))}{15 (d+e x)^{5/2} (b d-a e)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 74, normalized size = 0.8 \begin{align*} -{\frac{-4\,Abex+10\,Baex-6\,Bbdx+6\,Aae-10\,Abd+4\,Bad}{15\,{a}^{2}{e}^{2}-30\,bead+15\,{b}^{2}{d}^{2}} \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \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 [B] time = 16.5552, size = 460, normalized size = 4.84 \begin{align*} -\frac{2 \,{\left (3 \, A a^{2} e -{\left (3 \, B b^{2} d -{\left (5 \, B a b - 2 \, A b^{2}\right )} e\right )} x^{2} +{\left (2 \, B a^{2} - 5 \, A a b\right )} d -{\left ({\left (B a b + 5 \, A b^{2}\right )} d -{\left (5 \, B a^{2} + A a b\right )} e\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{15 \,{\left (b^{2} d^{5} - 2 \, a b d^{4} e + a^{2} d^{3} e^{2} +{\left (b^{2} d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}\right )} x^{3} + 3 \,{\left (b^{2} d^{3} e^{2} - 2 \, a b d^{2} e^{3} + a^{2} d e^{4}\right )} x^{2} + 3 \,{\left (b^{2} d^{4} e - 2 \, a b d^{3} e^{2} + a^{2} d^{2} e^{3}\right )} x\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 = 2.26562, size = 281, normalized size = 2.96 \begin{align*} -\frac{{\left (b x + a\right )}^{\frac{3}{2}}{\left (\frac{{\left (3 \, B b^{6} d{\left | b \right |} e^{2} - 5 \, B a b^{5}{\left | b \right |} e^{3} + 2 \, A b^{6}{\left | b \right |} e^{3}\right )}{\left (b x + a\right )}}{b^{12} d^{3} e^{6} - 3 \, a b^{11} d^{2} e^{7} + 3 \, a^{2} b^{10} d e^{8} - a^{3} b^{9} e^{9}} - \frac{5 \,{\left (B a b^{6} d{\left | b \right |} e^{2} - A b^{7} d{\left | b \right |} e^{2} - B a^{2} b^{5}{\left | b \right |} e^{3} + A a b^{6}{\left | b \right |} e^{3}\right )}}{b^{12} d^{3} e^{6} - 3 \, a b^{11} d^{2} e^{7} + 3 \, a^{2} b^{10} d e^{8} - a^{3} b^{9} e^{9}}\right )}}{960 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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