Optimal. Leaf size=95 \[ \frac{2 (a+b x)^{5/2} (-7 a B e+2 A b e+5 b B d)}{35 e (d+e x)^{5/2} (b d-a e)^2}-\frac{2 (a+b x)^{5/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)} \]
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Rubi [A] time = 0.051143, 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} \[ \frac{2 (a+b x)^{5/2} (-7 a B e+2 A b e+5 b B d)}{35 e (d+e x)^{5/2} (b d-a e)^2}-\frac{2 (a+b x)^{5/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 78
Rule 37
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2} (A+B x)}{(d+e x)^{9/2}} \, dx &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac{(5 b B d+2 A b e-7 a B e) \int \frac{(a+b x)^{3/2}}{(d+e x)^{7/2}} \, dx}{7 e (b d-a e)}\\ &=-\frac{2 (B d-A e) (a+b x)^{5/2}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac{2 (5 b B d+2 A b e-7 a B e) (a+b x)^{5/2}}{35 e (b d-a e)^2 (d+e x)^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0530805, size = 66, normalized size = 0.69 \[ \frac{2 (a+b x)^{5/2} (A (-5 a e+7 b d+2 b e x)+B (-2 a d-7 a e x+5 b d x))}{35 (d+e x)^{7/2} (b d-a e)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 74, normalized size = 0.8 \begin{align*} -{\frac{-4\,Abex+14\,Baex-10\,Bbdx+10\,Aae-14\,Abd+4\,Bad}{35\,{a}^{2}{e}^{2}-70\,bead+35\,{b}^{2}{d}^{2}} \left ( bx+a \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{7}{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 = 56.1361, size = 628, normalized size = 6.61 \begin{align*} -\frac{2 \,{\left (5 \, A a^{3} e -{\left (5 \, B b^{3} d -{\left (7 \, B a b^{2} - 2 \, A b^{3}\right )} e\right )} x^{3} -{\left ({\left (8 \, B a b^{2} + 7 \, A b^{3}\right )} d -{\left (14 \, B a^{2} b + A a b^{2}\right )} e\right )} x^{2} +{\left (2 \, B a^{3} - 7 \, A a^{2} b\right )} d -{\left ({\left (B a^{2} b + 14 \, A a b^{2}\right )} d -{\left (7 \, B a^{3} + 8 \, A a^{2} b\right )} e\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{35 \,{\left (b^{2} d^{6} - 2 \, a b d^{5} e + a^{2} d^{4} e^{2} +{\left (b^{2} d^{2} e^{4} - 2 \, a b d e^{5} + a^{2} e^{6}\right )} x^{4} + 4 \,{\left (b^{2} d^{3} e^{3} - 2 \, a b d^{2} e^{4} + a^{2} d e^{5}\right )} x^{3} + 6 \,{\left (b^{2} d^{4} e^{2} - 2 \, a b d^{3} e^{3} + a^{2} d^{2} e^{4}\right )} x^{2} + 4 \,{\left (b^{2} d^{5} e - 2 \, a b d^{4} e^{2} + a^{2} d^{3} 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 = 2.37289, size = 394, normalized size = 4.15 \begin{align*} -\frac{{\left (b x + a\right )}^{\frac{5}{2}}{\left (\frac{{\left (5 \, B b^{9} d^{2}{\left | b \right |} e^{3} - 12 \, B a b^{8} d{\left | b \right |} e^{4} + 2 \, A b^{9} d{\left | b \right |} e^{4} + 7 \, B a^{2} b^{7}{\left | b \right |} e^{5} - 2 \, A a b^{8}{\left | b \right |} e^{5}\right )}{\left (b x + a\right )}}{b^{16} d^{4} e^{8} - 4 \, a b^{15} d^{3} e^{9} + 6 \, a^{2} b^{14} d^{2} e^{10} - 4 \, a^{3} b^{13} d e^{11} + a^{4} b^{12} e^{12}} - \frac{7 \,{\left (B a b^{9} d^{2}{\left | b \right |} e^{3} - A b^{10} d^{2}{\left | b \right |} e^{3} - 2 \, B a^{2} b^{8} d{\left | b \right |} e^{4} + 2 \, A a b^{9} d{\left | b \right |} e^{4} + B a^{3} b^{7}{\left | b \right |} e^{5} - A a^{2} b^{8}{\left | b \right |} e^{5}\right )}}{b^{16} d^{4} e^{8} - 4 \, a b^{15} d^{3} e^{9} + 6 \, a^{2} b^{14} d^{2} e^{10} - 4 \, a^{3} b^{13} d e^{11} + a^{4} b^{12} e^{12}}\right )}}{26880 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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