Optimal. Leaf size=147 \[ \frac {4 b (a+b x)^{5/2} (-9 a B e+4 A b e+5 b B d)}{315 e (d+e x)^{5/2} (b d-a e)^3}+\frac {2 (a+b x)^{5/2} (-9 a B e+4 A b e+5 b B d)}{63 e (d+e x)^{7/2} (b d-a e)^2}-\frac {2 (a+b x)^{5/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \]
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Rubi [A] time = 0.09, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {78, 45, 37} \[ \frac {4 b (a+b x)^{5/2} (-9 a B e+4 A b e+5 b B d)}{315 e (d+e x)^{5/2} (b d-a e)^3}+\frac {2 (a+b x)^{5/2} (-9 a B e+4 A b e+5 b B d)}{63 e (d+e x)^{7/2} (b d-a e)^2}-\frac {2 (a+b x)^{5/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{11/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {(5 b B d+4 A b e-9 a B e) \int \frac {(a+b x)^{3/2}}{(d+e x)^{9/2}} \, dx}{9 e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {2 (5 b B d+4 A b e-9 a B e) (a+b x)^{5/2}}{63 e (b d-a e)^2 (d+e x)^{7/2}}+\frac {(2 b (5 b B d+4 A b e-9 a B e)) \int \frac {(a+b x)^{3/2}}{(d+e x)^{7/2}} \, dx}{63 e (b d-a e)^2}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {2 (5 b B d+4 A b e-9 a B e) (a+b x)^{5/2}}{63 e (b d-a e)^2 (d+e x)^{7/2}}+\frac {4 b (5 b B d+4 A b e-9 a B e) (a+b x)^{5/2}}{315 e (b d-a e)^3 (d+e x)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 135, normalized size = 0.92 \[ \frac {2 (a+b x)^{5/2} \left (A \left (35 a^2 e^2-10 a b e (9 d+2 e x)+b^2 \left (63 d^2+36 d e x+8 e^2 x^2\right )\right )+B \left (5 a^2 e (2 d+9 e x)-2 a b \left (9 d^2+53 d e x+9 e^2 x^2\right )+5 b^2 d x (9 d+2 e x)\right )\right )}{315 (d+e x)^{9/2} (b d-a e)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 110.90, size = 567, normalized size = 3.86 \[ \frac {2 \, {\left (35 \, A a^{4} e^{2} + 2 \, {\left (5 \, B b^{4} d e - {\left (9 \, B a b^{3} - 4 \, A b^{4}\right )} e^{2}\right )} x^{4} + {\left (45 \, B b^{4} d^{2} - 2 \, {\left (43 \, B a b^{3} - 18 \, A b^{4}\right )} d e + {\left (9 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} e^{2}\right )} x^{3} - 9 \, {\left (2 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} d^{2} + 10 \, {\left (B a^{4} - 9 \, A a^{3} b\right )} d e + 3 \, {\left (3 \, {\left (8 \, B a b^{3} + 7 \, A b^{4}\right )} d^{2} - 2 \, {\left (32 \, B a^{2} b^{2} + 3 \, A a b^{3}\right )} d e + {\left (24 \, B a^{3} b + A a^{2} b^{2}\right )} e^{2}\right )} x^{2} + {\left (9 \, {\left (B a^{2} b^{2} + 14 \, A a b^{3}\right )} d^{2} - 2 \, {\left (43 \, B a^{3} b + 72 \, A a^{2} b^{2}\right )} d e + 5 \, {\left (9 \, B a^{4} + 10 \, A a^{3} b\right )} e^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{315 \, {\left (b^{3} d^{8} - 3 \, a b^{2} d^{7} e + 3 \, a^{2} b d^{6} e^{2} - a^{3} d^{5} e^{3} + {\left (b^{3} d^{3} e^{5} - 3 \, a b^{2} d^{2} e^{6} + 3 \, a^{2} b d e^{7} - a^{3} e^{8}\right )} x^{5} + 5 \, {\left (b^{3} d^{4} e^{4} - 3 \, a b^{2} d^{3} e^{5} + 3 \, a^{2} b d^{2} e^{6} - a^{3} d e^{7}\right )} x^{4} + 10 \, {\left (b^{3} d^{5} e^{3} - 3 \, a b^{2} d^{4} e^{4} + 3 \, a^{2} b d^{3} e^{5} - a^{3} d^{2} e^{6}\right )} x^{3} + 10 \, {\left (b^{3} d^{6} e^{2} - 3 \, a b^{2} d^{5} e^{3} + 3 \, a^{2} b d^{4} e^{4} - a^{3} d^{3} e^{5}\right )} x^{2} + 5 \, {\left (b^{3} d^{7} e - 3 \, a b^{2} d^{6} e^{2} + 3 \, a^{2} b d^{5} e^{3} - a^{3} d^{4} e^{4}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.55, size = 483, normalized size = 3.29 \[ \frac {2 \, {\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (5 \, B b^{11} d^{2} {\left | b \right |} e^{5} - 14 \, B a b^{10} d {\left | b \right |} e^{6} + 4 \, A b^{11} d {\left | b \right |} e^{6} + 9 \, B a^{2} b^{9} {\left | b \right |} e^{7} - 4 \, A a b^{10} {\left | b \right |} e^{7}\right )} {\left (b x + a\right )}}{b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}} + \frac {9 \, {\left (5 \, B b^{12} d^{3} {\left | b \right |} e^{4} - 19 \, B a b^{11} d^{2} {\left | b \right |} e^{5} + 4 \, A b^{12} d^{2} {\left | b \right |} e^{5} + 23 \, B a^{2} b^{10} d {\left | b \right |} e^{6} - 8 \, A a b^{11} d {\left | b \right |} e^{6} - 9 \, B a^{3} b^{9} {\left | b \right |} e^{7} + 4 \, A a^{2} b^{10} {\left | b \right |} e^{7}\right )}}{b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}}\right )} - \frac {63 \, {\left (B a b^{12} d^{3} {\left | b \right |} e^{4} - A b^{13} d^{3} {\left | b \right |} e^{4} - 3 \, B a^{2} b^{11} d^{2} {\left | b \right |} e^{5} + 3 \, A a b^{12} d^{2} {\left | b \right |} e^{5} + 3 \, B a^{3} b^{10} d {\left | b \right |} e^{6} - 3 \, A a^{2} b^{11} d {\left | b \right |} e^{6} - B a^{4} b^{9} {\left | b \right |} e^{7} + A a^{3} b^{10} {\left | b \right |} e^{7}\right )}}{b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}}\right )} {\left (b x + a\right )}^{\frac {5}{2}}}{315 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 177, normalized size = 1.20 \[ -\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (8 A \,b^{2} e^{2} x^{2}-18 B a b \,e^{2} x^{2}+10 B \,b^{2} d e \,x^{2}-20 A a b \,e^{2} x +36 A \,b^{2} d e x +45 B \,a^{2} e^{2} x -106 B a b d e x +45 B \,b^{2} d^{2} x +35 A \,a^{2} e^{2}-90 A a b d e +63 A \,b^{2} d^{2}+10 B \,a^{2} d e -18 B a b \,d^{2}\right )}{315 \left (e x +d \right )^{\frac {9}{2}} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.37, size = 402, normalized size = 2.73 \[ -\frac {\sqrt {d+e\,x}\,\left (\frac {\sqrt {a+b\,x}\,\left (20\,B\,a^4\,d\,e+70\,A\,a^4\,e^2-36\,B\,a^3\,b\,d^2-180\,A\,a^3\,b\,d\,e+126\,A\,a^2\,b^2\,d^2\right )}{315\,e^5\,{\left (a\,e-b\,d\right )}^3}+\frac {x^2\,\sqrt {a+b\,x}\,\left (144\,B\,a^3\,b\,e^2-384\,B\,a^2\,b^2\,d\,e+6\,A\,a^2\,b^2\,e^2+144\,B\,a\,b^3\,d^2-36\,A\,a\,b^3\,d\,e+126\,A\,b^4\,d^2\right )}{315\,e^5\,{\left (a\,e-b\,d\right )}^3}+\frac {x\,\sqrt {a+b\,x}\,\left (90\,B\,a^4\,e^2-172\,B\,a^3\,b\,d\,e+100\,A\,a^3\,b\,e^2+18\,B\,a^2\,b^2\,d^2-288\,A\,a^2\,b^2\,d\,e+252\,A\,a\,b^3\,d^2\right )}{315\,e^5\,{\left (a\,e-b\,d\right )}^3}+\frac {4\,b^3\,x^4\,\sqrt {a+b\,x}\,\left (4\,A\,b\,e-9\,B\,a\,e+5\,B\,b\,d\right )}{315\,e^4\,{\left (a\,e-b\,d\right )}^3}-\frac {2\,b^2\,x^3\,\left (a\,e-9\,b\,d\right )\,\sqrt {a+b\,x}\,\left (4\,A\,b\,e-9\,B\,a\,e+5\,B\,b\,d\right )}{315\,e^5\,{\left (a\,e-b\,d\right )}^3}\right )}{x^5+\frac {d^5}{e^5}+\frac {5\,d\,x^4}{e}+\frac {5\,d^4\,x}{e^4}+\frac {10\,d^2\,x^3}{e^2}+\frac {10\,d^3\,x^2}{e^3}} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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