Optimal. Leaf size=95 \[ \frac {2 (a+b x)^{7/2} (-9 a B e+2 A b e+7 b B d)}{63 e (d+e x)^{7/2} (b d-a e)^2}-\frac {2 (a+b x)^{7/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \]
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Rubi [A] time = 0.05, antiderivative size = 95, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {2 (a+b x)^{7/2} (-9 a B e+2 A b e+7 b B d)}{63 e (d+e x)^{7/2} (b d-a e)^2}-\frac {2 (a+b x)^{7/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{11/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {(7 b B d+2 A b e-9 a B e) \int \frac {(a+b x)^{5/2}}{(d+e x)^{9/2}} \, dx}{9 e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {2 (7 b B d+2 A b e-9 a B e) (a+b x)^{7/2}}{63 e (b d-a e)^2 (d+e x)^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 66, normalized size = 0.69 \begin {gather*} \frac {2 (a+b x)^{7/2} (A (-7 a e+9 b d+2 b e x)+B (-2 a d-9 a e x+7 b d x))}{63 (d+e x)^{9/2} (b d-a e)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 73, normalized size = 0.77 \begin {gather*} -\frac {2 (a+b x)^{9/2} \left (-\frac {9 A b (d+e x)}{a+b x}+\frac {9 a B (d+e x)}{a+b x}+7 A e-7 B d\right )}{63 (d+e x)^{9/2} (b d-a e)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 152.73, size = 391, normalized size = 4.12 \begin {gather*} -\frac {2 \, {\left (7 \, A a^{4} e - {\left (7 \, B b^{4} d - {\left (9 \, B a b^{3} - 2 \, A b^{4}\right )} e\right )} x^{4} - {\left ({\left (19 \, B a b^{3} + 9 \, A b^{4}\right )} d - {\left (27 \, B a^{2} b^{2} + A a b^{3}\right )} e\right )} x^{3} - 3 \, {\left ({\left (5 \, B a^{2} b^{2} + 9 \, A a b^{3}\right )} d - {\left (9 \, B a^{3} b + 5 \, A a^{2} b^{2}\right )} e\right )} x^{2} + {\left (2 \, B a^{4} - 9 \, A a^{3} b\right )} d - {\left ({\left (B a^{3} b + 27 \, A a^{2} b^{2}\right )} d - {\left (9 \, B a^{4} + 19 \, A a^{3} b\right )} e\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{63 \, {\left (b^{2} d^{7} - 2 \, a b d^{6} e + a^{2} d^{5} e^{2} + {\left (b^{2} d^{2} e^{5} - 2 \, a b d e^{6} + a^{2} e^{7}\right )} x^{5} + 5 \, {\left (b^{2} d^{3} e^{4} - 2 \, a b d^{2} e^{5} + a^{2} d e^{6}\right )} x^{4} + 10 \, {\left (b^{2} d^{4} e^{3} - 2 \, a b d^{3} e^{4} + a^{2} d^{2} e^{5}\right )} x^{3} + 10 \, {\left (b^{2} d^{5} e^{2} - 2 \, a b d^{4} e^{3} + a^{2} d^{3} e^{4}\right )} x^{2} + 5 \, {\left (b^{2} d^{6} e - 2 \, a b d^{5} e^{2} + a^{2} d^{4} e^{3}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 5.77, size = 356, normalized size = 3.75 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {7}{2}} {\left (\frac {{\left (7 \, B b^{12} d^{3} {\left | b \right |} e^{4} - 23 \, B a b^{11} d^{2} {\left | b \right |} e^{5} + 2 \, A b^{12} d^{2} {\left | b \right |} e^{5} + 25 \, B a^{2} b^{10} d {\left | b \right |} e^{6} - 4 \, A a b^{11} d {\left | b \right |} e^{6} - 9 \, B a^{3} b^{9} {\left | b \right |} e^{7} + 2 \, A a^{2} b^{10} {\left | b \right |} e^{7}\right )} {\left (b x + a\right )}}{b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}} - \frac {9 \, {\left (B a b^{12} d^{3} {\left | b \right |} e^{4} - A b^{13} d^{3} {\left | b \right |} e^{4} - 3 \, B a^{2} b^{11} d^{2} {\left | b \right |} e^{5} + 3 \, A a b^{12} d^{2} {\left | b \right |} e^{5} + 3 \, B a^{3} b^{10} d {\left | b \right |} e^{6} - 3 \, A a^{2} b^{11} d {\left | b \right |} e^{6} - B a^{4} b^{9} {\left | b \right |} e^{7} + A a^{3} b^{10} {\left | b \right |} e^{7}\right )}}{b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}}\right )}}{63 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 74, normalized size = 0.78 \begin {gather*} -\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-2 A b e x +9 B a e x -7 B b d x +7 A a e -9 A b d +2 B a d \right )}{63 \left (e x +d \right )^{\frac {9}{2}} \left (a^{2} e^{2}-2 b e a d +b^{2} d^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.40, size = 325, normalized size = 3.42 \begin {gather*} \frac {\sqrt {d+e\,x}\,\left (\frac {x^3\,\sqrt {a+b\,x}\,\left (18\,A\,b^4\,d-2\,A\,a\,b^3\,e+38\,B\,a\,b^3\,d-54\,B\,a^2\,b^2\,e\right )}{63\,e^5\,{\left (a\,e-b\,d\right )}^2}-\frac {\sqrt {a+b\,x}\,\left (14\,A\,a^4\,e+4\,B\,a^4\,d-18\,A\,a^3\,b\,d\right )}{63\,e^5\,{\left (a\,e-b\,d\right )}^2}+\frac {x^4\,\sqrt {a+b\,x}\,\left (4\,A\,b^4\,e+14\,B\,b^4\,d-18\,B\,a\,b^3\,e\right )}{63\,e^5\,{\left (a\,e-b\,d\right )}^2}-\frac {x\,\sqrt {a+b\,x}\,\left (18\,B\,a^4\,e+38\,A\,a^3\,b\,e-2\,B\,a^3\,b\,d-54\,A\,a^2\,b^2\,d\right )}{63\,e^5\,{\left (a\,e-b\,d\right )}^2}+\frac {2\,a\,b\,x^2\,\sqrt {a+b\,x}\,\left (9\,A\,b^2\,d-9\,B\,a^2\,e-5\,A\,a\,b\,e+5\,B\,a\,b\,d\right )}{21\,e^5\,{\left (a\,e-b\,d\right )}^2}\right )}{x^5+\frac {d^5}{e^5}+\frac {5\,d\,x^4}{e}+\frac {5\,d^4\,x}{e^4}+\frac {10\,d^2\,x^3}{e^2}+\frac {10\,d^3\,x^2}{e^3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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