Optimal. Leaf size=246 \[ \frac {5 (b d-a e)^3 (a B e-8 A b e+7 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{3/2} e^{9/2}}-\frac {5 \sqrt {a+b x} \sqrt {d+e x} (b d-a e)^2 (a B e-8 A b e+7 b B d)}{64 b e^4}+\frac {5 (a+b x)^{3/2} \sqrt {d+e x} (b d-a e) (a B e-8 A b e+7 b B d)}{96 b e^3}-\frac {(a+b x)^{5/2} \sqrt {d+e x} (a B e-8 A b e+7 b B d)}{24 b e^2}+\frac {B (a+b x)^{7/2} \sqrt {d+e x}}{4 b e} \]
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Rubi [A] time = 0.20, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {80, 50, 63, 217, 206} \[ \frac {5 (b d-a e)^3 (a B e-8 A b e+7 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{3/2} e^{9/2}}-\frac {(a+b x)^{5/2} \sqrt {d+e x} (a B e-8 A b e+7 b B d)}{24 b e^2}+\frac {5 (a+b x)^{3/2} \sqrt {d+e x} (b d-a e) (a B e-8 A b e+7 b B d)}{96 b e^3}-\frac {5 \sqrt {a+b x} \sqrt {d+e x} (b d-a e)^2 (a B e-8 A b e+7 b B d)}{64 b e^4}+\frac {B (a+b x)^{7/2} \sqrt {d+e x}}{4 b e} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2} (A+B x)}{\sqrt {d+e x}} \, dx &=\frac {B (a+b x)^{7/2} \sqrt {d+e x}}{4 b e}+\frac {\left (4 A b e-B \left (\frac {7 b d}{2}+\frac {a e}{2}\right )\right ) \int \frac {(a+b x)^{5/2}}{\sqrt {d+e x}} \, dx}{4 b e}\\ &=-\frac {(7 b B d-8 A b e+a B e) (a+b x)^{5/2} \sqrt {d+e x}}{24 b e^2}+\frac {B (a+b x)^{7/2} \sqrt {d+e x}}{4 b e}+\frac {(5 (b d-a e) (7 b B d-8 A b e+a B e)) \int \frac {(a+b x)^{3/2}}{\sqrt {d+e x}} \, dx}{48 b e^2}\\ &=\frac {5 (b d-a e) (7 b B d-8 A b e+a B e) (a+b x)^{3/2} \sqrt {d+e x}}{96 b e^3}-\frac {(7 b B d-8 A b e+a B e) (a+b x)^{5/2} \sqrt {d+e x}}{24 b e^2}+\frac {B (a+b x)^{7/2} \sqrt {d+e x}}{4 b e}-\frac {\left (5 (b d-a e)^2 (7 b B d-8 A b e+a B e)\right ) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}} \, dx}{64 b e^3}\\ &=-\frac {5 (b d-a e)^2 (7 b B d-8 A b e+a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b e^4}+\frac {5 (b d-a e) (7 b B d-8 A b e+a B e) (a+b x)^{3/2} \sqrt {d+e x}}{96 b e^3}-\frac {(7 b B d-8 A b e+a B e) (a+b x)^{5/2} \sqrt {d+e x}}{24 b e^2}+\frac {B (a+b x)^{7/2} \sqrt {d+e x}}{4 b e}+\frac {\left (5 (b d-a e)^3 (7 b B d-8 A b e+a B e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{128 b e^4}\\ &=-\frac {5 (b d-a e)^2 (7 b B d-8 A b e+a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b e^4}+\frac {5 (b d-a e) (7 b B d-8 A b e+a B e) (a+b x)^{3/2} \sqrt {d+e x}}{96 b e^3}-\frac {(7 b B d-8 A b e+a B e) (a+b x)^{5/2} \sqrt {d+e x}}{24 b e^2}+\frac {B (a+b x)^{7/2} \sqrt {d+e x}}{4 b e}+\frac {\left (5 (b d-a e)^3 (7 b B d-8 A b e+a B e)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{64 b^2 e^4}\\ &=-\frac {5 (b d-a e)^2 (7 b B d-8 A b e+a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b e^4}+\frac {5 (b d-a e) (7 b B d-8 A b e+a B e) (a+b x)^{3/2} \sqrt {d+e x}}{96 b e^3}-\frac {(7 b B d-8 A b e+a B e) (a+b x)^{5/2} \sqrt {d+e x}}{24 b e^2}+\frac {B (a+b x)^{7/2} \sqrt {d+e x}}{4 b e}+\frac {\left (5 (b d-a e)^3 (7 b B d-8 A b e+a B e)\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{64 b^2 e^4}\\ &=-\frac {5 (b d-a e)^2 (7 b B d-8 A b e+a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b e^4}+\frac {5 (b d-a e) (7 b B d-8 A b e+a B e) (a+b x)^{3/2} \sqrt {d+e x}}{96 b e^3}-\frac {(7 b B d-8 A b e+a B e) (a+b x)^{5/2} \sqrt {d+e x}}{24 b e^2}+\frac {B (a+b x)^{7/2} \sqrt {d+e x}}{4 b e}+\frac {5 (b d-a e)^3 (7 b B d-8 A b e+a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{3/2} e^{9/2}}\\ \end {align*}
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Mathematica [A] time = 1.47, size = 297, normalized size = 1.21 \[ \frac {\sqrt {d+e x} \left (\frac {b (-a B e+8 A b e-7 b B d) \left (8 b^3 e^3 (a+b x)^3 \sqrt {b d-a e} \sqrt {\frac {b (d+e x)}{b d-a e}}-10 b^3 e^2 (a+b x)^2 (b d-a e)^{3/2} \sqrt {\frac {b (d+e x)}{b d-a e}}+15 b^3 e (a+b x) (b d-a e)^{5/2} \sqrt {\frac {b (d+e x)}{b d-a e}}-15 b^3 \sqrt {e} \sqrt {a+b x} (b d-a e)^3 \sinh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b d-a e}}\right )\right )}{3 \sqrt {b d-a e} \sqrt {\frac {b (d+e x)}{b d-a e}}}+16 b^4 B e^4 (a+b x)^4\right )}{64 b^5 e^5 \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 770, normalized size = 3.13 \[ \left [-\frac {15 \, {\left (7 \, B b^{4} d^{4} - 4 \, {\left (5 \, B a b^{3} + 2 \, A b^{4}\right )} d^{3} e + 6 \, {\left (3 \, B a^{2} b^{2} + 4 \, A a b^{3}\right )} d^{2} e^{2} - 4 \, {\left (B a^{3} b + 6 \, A a^{2} b^{2}\right )} d e^{3} - {\left (B a^{4} - 8 \, A a^{3} b\right )} e^{4}\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} - 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) - 4 \, {\left (48 \, B b^{4} e^{4} x^{3} - 105 \, B b^{4} d^{3} e + 5 \, {\left (53 \, B a b^{3} + 24 \, A b^{4}\right )} d^{2} e^{2} - {\left (191 \, B a^{2} b^{2} + 320 \, A a b^{3}\right )} d e^{3} + 3 \, {\left (5 \, B a^{3} b + 88 \, A a^{2} b^{2}\right )} e^{4} - 8 \, {\left (7 \, B b^{4} d e^{3} - {\left (17 \, B a b^{3} + 8 \, A b^{4}\right )} e^{4}\right )} x^{2} + 2 \, {\left (35 \, B b^{4} d^{2} e^{2} - 2 \, {\left (43 \, B a b^{3} + 20 \, A b^{4}\right )} d e^{3} + {\left (59 \, B a^{2} b^{2} + 104 \, A a b^{3}\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{768 \, b^{2} e^{5}}, -\frac {15 \, {\left (7 \, B b^{4} d^{4} - 4 \, {\left (5 \, B a b^{3} + 2 \, A b^{4}\right )} d^{3} e + 6 \, {\left (3 \, B a^{2} b^{2} + 4 \, A a b^{3}\right )} d^{2} e^{2} - 4 \, {\left (B a^{3} b + 6 \, A a^{2} b^{2}\right )} d e^{3} - {\left (B a^{4} - 8 \, A a^{3} b\right )} e^{4}\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) - 2 \, {\left (48 \, B b^{4} e^{4} x^{3} - 105 \, B b^{4} d^{3} e + 5 \, {\left (53 \, B a b^{3} + 24 \, A b^{4}\right )} d^{2} e^{2} - {\left (191 \, B a^{2} b^{2} + 320 \, A a b^{3}\right )} d e^{3} + 3 \, {\left (5 \, B a^{3} b + 88 \, A a^{2} b^{2}\right )} e^{4} - 8 \, {\left (7 \, B b^{4} d e^{3} - {\left (17 \, B a b^{3} + 8 \, A b^{4}\right )} e^{4}\right )} x^{2} + 2 \, {\left (35 \, B b^{4} d^{2} e^{2} - 2 \, {\left (43 \, B a b^{3} + 20 \, A b^{4}\right )} d e^{3} + {\left (59 \, B a^{2} b^{2} + 104 \, A a b^{3}\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{384 \, b^{2} e^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.63, size = 390, normalized size = 1.59 \[ \frac {{\left (\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )} B e^{\left (-1\right )}}{b^{2}} - \frac {{\left (7 \, B b^{3} d e^{5} + B a b^{2} e^{6} - 8 \, A b^{3} e^{6}\right )} e^{\left (-7\right )}}{b^{4}}\right )} + \frac {5 \, {\left (7 \, B b^{4} d^{2} e^{4} - 6 \, B a b^{3} d e^{5} - 8 \, A b^{4} d e^{5} - B a^{2} b^{2} e^{6} + 8 \, A a b^{3} e^{6}\right )} e^{\left (-7\right )}}{b^{4}}\right )} - \frac {15 \, {\left (7 \, B b^{5} d^{3} e^{3} - 13 \, B a b^{4} d^{2} e^{4} - 8 \, A b^{5} d^{2} e^{4} + 5 \, B a^{2} b^{3} d e^{5} + 16 \, A a b^{4} d e^{5} + B a^{3} b^{2} e^{6} - 8 \, A a^{2} b^{3} e^{6}\right )} e^{\left (-7\right )}}{b^{4}}\right )} \sqrt {b x + a} - \frac {15 \, {\left (7 \, B b^{4} d^{4} - 20 \, B a b^{3} d^{3} e - 8 \, A b^{4} d^{3} e + 18 \, B a^{2} b^{2} d^{2} e^{2} + 24 \, A a b^{3} d^{2} e^{2} - 4 \, B a^{3} b d e^{3} - 24 \, A a^{2} b^{2} d e^{3} - B a^{4} e^{4} + 8 \, A a^{3} b e^{4}\right )} e^{\left (-\frac {9}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{b^{\frac {3}{2}}}\right )} b}{192 \, {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 968, normalized size = 3.93 \[ \frac {\sqrt {b x +a}\, \sqrt {e x +d}\, \left (120 A \,a^{3} b \,e^{4} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-360 A \,a^{2} b^{2} d \,e^{3} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+360 A a \,b^{3} d^{2} e^{2} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-120 A \,b^{4} d^{3} e \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-15 B \,a^{4} e^{4} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-60 B \,a^{3} b d \,e^{3} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+270 B \,a^{2} b^{2} d^{2} e^{2} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-300 B a \,b^{3} d^{3} e \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+105 B \,b^{4} d^{4} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+96 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, B \,b^{3} e^{3} x^{3}+128 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, A \,b^{3} e^{3} x^{2}+272 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, B a \,b^{2} e^{3} x^{2}-112 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, B \,b^{3} d \,e^{2} x^{2}+416 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, A a \,b^{2} e^{3} x -160 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, A \,b^{3} d \,e^{2} x +236 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, B \,a^{2} b \,e^{3} x -344 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, B a \,b^{2} d \,e^{2} x +140 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, B \,b^{3} d^{2} e x +528 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, A \,a^{2} b \,e^{3}-640 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, A a \,b^{2} d \,e^{2}+240 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, A \,b^{3} d^{2} e +30 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, B \,a^{3} e^{3}-382 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, B \,a^{2} b d \,e^{2}+530 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, B a \,b^{2} d^{2} e -210 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, B \,b^{3} d^{3}\right )}{384 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b \,e^{4}} \]
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 [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{5/2}}{\sqrt {d+e\,x}} \,d x \]
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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