Optimal. Leaf size=128 \[ -\frac {2 b (d+e x)^{11/2} (-2 a B e-A b e+3 b B d)}{11 e^4}+\frac {2 (d+e x)^{9/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{9 e^4}-\frac {2 (d+e x)^{7/2} (b d-a e)^2 (B d-A e)}{7 e^4}+\frac {2 b^2 B (d+e x)^{13/2}}{13 e^4} \]
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Rubi [A] time = 0.05, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {77} \[ -\frac {2 b (d+e x)^{11/2} (-2 a B e-A b e+3 b B d)}{11 e^4}+\frac {2 (d+e x)^{9/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{9 e^4}-\frac {2 (d+e x)^{7/2} (b d-a e)^2 (B d-A e)}{7 e^4}+\frac {2 b^2 B (d+e x)^{13/2}}{13 e^4} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int (a+b x)^2 (A+B x) (d+e x)^{5/2} \, dx &=\int \left (\frac {(-b d+a e)^2 (-B d+A e) (d+e x)^{5/2}}{e^3}+\frac {(-b d+a e) (-3 b B d+2 A b e+a B e) (d+e x)^{7/2}}{e^3}+\frac {b (-3 b B d+A b e+2 a B e) (d+e x)^{9/2}}{e^3}+\frac {b^2 B (d+e x)^{11/2}}{e^3}\right ) \, dx\\ &=-\frac {2 (b d-a e)^2 (B d-A e) (d+e x)^{7/2}}{7 e^4}+\frac {2 (b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{9/2}}{9 e^4}-\frac {2 b (3 b B d-A b e-2 a B e) (d+e x)^{11/2}}{11 e^4}+\frac {2 b^2 B (d+e x)^{13/2}}{13 e^4}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 107, normalized size = 0.84 \[ \frac {2 (d+e x)^{7/2} \left (-819 b (d+e x)^2 (-2 a B e-A b e+3 b B d)+1001 (d+e x) (b d-a e) (-a B e-2 A b e+3 b B d)-1287 (b d-a e)^2 (B d-A e)+693 b^2 B (d+e x)^3\right )}{9009 e^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.98, size = 356, normalized size = 2.78 \[ \frac {2 \, {\left (693 \, B b^{2} e^{6} x^{6} - 48 \, B b^{2} d^{6} + 1287 \, A a^{2} d^{3} e^{3} + 104 \, {\left (2 \, B a b + A b^{2}\right )} d^{5} e - 286 \, {\left (B a^{2} + 2 \, A a b\right )} d^{4} e^{2} + 63 \, {\left (27 \, B b^{2} d e^{5} + 13 \, {\left (2 \, B a b + A b^{2}\right )} e^{6}\right )} x^{5} + 7 \, {\left (159 \, B b^{2} d^{2} e^{4} + 299 \, {\left (2 \, B a b + A b^{2}\right )} d e^{5} + 143 \, {\left (B a^{2} + 2 \, A a b\right )} e^{6}\right )} x^{4} + {\left (15 \, B b^{2} d^{3} e^{3} + 1287 \, A a^{2} e^{6} + 1469 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e^{4} + 2717 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{5}\right )} x^{3} - 3 \, {\left (6 \, B b^{2} d^{4} e^{2} - 1287 \, A a^{2} d e^{5} - 13 \, {\left (2 \, B a b + A b^{2}\right )} d^{3} e^{3} - 715 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{4}\right )} x^{2} + {\left (24 \, B b^{2} d^{5} e + 3861 \, A a^{2} d^{2} e^{4} - 52 \, {\left (2 \, B a b + A b^{2}\right )} d^{4} e^{2} + 143 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} e^{3}\right )} x\right )} \sqrt {e x + d}}{9009 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.43, size = 1368, normalized size = 10.69 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 169, normalized size = 1.32 \[ \frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (693 b^{2} B \,x^{3} e^{3}+819 A \,b^{2} e^{3} x^{2}+1638 B a b \,e^{3} x^{2}-378 B \,b^{2} d \,e^{2} x^{2}+2002 A a b \,e^{3} x -364 A \,b^{2} d \,e^{2} x +1001 B \,a^{2} e^{3} x -728 B a b d \,e^{2} x +168 B \,b^{2} d^{2} e x +1287 a^{2} A \,e^{3}-572 A a b d \,e^{2}+104 A \,b^{2} d^{2} e -286 B \,a^{2} d \,e^{2}+208 B a b \,d^{2} e -48 B \,b^{2} d^{3}\right )}{9009 e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 159, normalized size = 1.24 \[ \frac {2 \, {\left (693 \, {\left (e x + d\right )}^{\frac {13}{2}} B b^{2} - 819 \, {\left (3 \, B b^{2} d - {\left (2 \, B a b + A b^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 1001 \, {\left (3 \, B b^{2} d^{2} - 2 \, {\left (2 \, B a b + A b^{2}\right )} d e + {\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 1287 \, {\left (B b^{2} d^{3} - A a^{2} e^{3} - {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{9009 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 115, normalized size = 0.90 \[ \frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,b^2\,e-6\,B\,b^2\,d+4\,B\,a\,b\,e\right )}{11\,e^4}+\frac {2\,B\,b^2\,{\left (d+e\,x\right )}^{13/2}}{13\,e^4}+\frac {2\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,b\,e+B\,a\,e-3\,B\,b\,d\right )}{9\,e^4}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.92, size = 857, normalized size = 6.70 \[ \begin {cases} \frac {2 A a^{2} d^{3} \sqrt {d + e x}}{7 e} + \frac {6 A a^{2} d^{2} x \sqrt {d + e x}}{7} + \frac {6 A a^{2} d e x^{2} \sqrt {d + e x}}{7} + \frac {2 A a^{2} e^{2} x^{3} \sqrt {d + e x}}{7} - \frac {8 A a b d^{4} \sqrt {d + e x}}{63 e^{2}} + \frac {4 A a b d^{3} x \sqrt {d + e x}}{63 e} + \frac {20 A a b d^{2} x^{2} \sqrt {d + e x}}{21} + \frac {76 A a b d e x^{3} \sqrt {d + e x}}{63} + \frac {4 A a b e^{2} x^{4} \sqrt {d + e x}}{9} + \frac {16 A b^{2} d^{5} \sqrt {d + e x}}{693 e^{3}} - \frac {8 A b^{2} d^{4} x \sqrt {d + e x}}{693 e^{2}} + \frac {2 A b^{2} d^{3} x^{2} \sqrt {d + e x}}{231 e} + \frac {226 A b^{2} d^{2} x^{3} \sqrt {d + e x}}{693} + \frac {46 A b^{2} d e x^{4} \sqrt {d + e x}}{99} + \frac {2 A b^{2} e^{2} x^{5} \sqrt {d + e x}}{11} - \frac {4 B a^{2} d^{4} \sqrt {d + e x}}{63 e^{2}} + \frac {2 B a^{2} d^{3} x \sqrt {d + e x}}{63 e} + \frac {10 B a^{2} d^{2} x^{2} \sqrt {d + e x}}{21} + \frac {38 B a^{2} d e x^{3} \sqrt {d + e x}}{63} + \frac {2 B a^{2} e^{2} x^{4} \sqrt {d + e x}}{9} + \frac {32 B a b d^{5} \sqrt {d + e x}}{693 e^{3}} - \frac {16 B a b d^{4} x \sqrt {d + e x}}{693 e^{2}} + \frac {4 B a b d^{3} x^{2} \sqrt {d + e x}}{231 e} + \frac {452 B a b d^{2} x^{3} \sqrt {d + e x}}{693} + \frac {92 B a b d e x^{4} \sqrt {d + e x}}{99} + \frac {4 B a b e^{2} x^{5} \sqrt {d + e x}}{11} - \frac {32 B b^{2} d^{6} \sqrt {d + e x}}{3003 e^{4}} + \frac {16 B b^{2} d^{5} x \sqrt {d + e x}}{3003 e^{3}} - \frac {4 B b^{2} d^{4} x^{2} \sqrt {d + e x}}{1001 e^{2}} + \frac {10 B b^{2} d^{3} x^{3} \sqrt {d + e x}}{3003 e} + \frac {106 B b^{2} d^{2} x^{4} \sqrt {d + e x}}{429} + \frac {54 B b^{2} d e x^{5} \sqrt {d + e x}}{143} + \frac {2 B b^{2} e^{2} x^{6} \sqrt {d + e x}}{13} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (A a^{2} x + A a b x^{2} + \frac {A b^{2} x^{3}}{3} + \frac {B a^{2} x^{2}}{2} + \frac {2 B a b x^{3}}{3} + \frac {B b^{2} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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