Optimal. Leaf size=193 \[ -\frac {(b d-a e)^2 (5 a B e-6 A b e+b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{7/2} e^{3/2}}-\frac {\sqrt {a+b x} \sqrt {d+e x} (b d-a e) (5 a B e-6 A b e+b B d)}{8 b^3 e}-\frac {\sqrt {a+b x} (d+e x)^{3/2} (5 a B e-6 A b e+b B d)}{12 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{5/2}}{3 b e} \]
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Rubi [A] time = 0.15, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {(b d-a e)^2 (5 a B e-6 A b e+b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{7/2} e^{3/2}}-\frac {\sqrt {a+b x} (d+e x)^{3/2} (5 a B e-6 A b e+b B d)}{12 b^2 e}-\frac {\sqrt {a+b x} \sqrt {d+e x} (b d-a e) (5 a B e-6 A b e+b B d)}{8 b^3 e}+\frac {B \sqrt {a+b x} (d+e x)^{5/2}}{3 b e} \]
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) (d+e x)^{3/2}}{\sqrt {a+b x}} \, dx &=\frac {B \sqrt {a+b x} (d+e x)^{5/2}}{3 b e}+\frac {\left (3 A b e-B \left (\frac {b d}{2}+\frac {5 a e}{2}\right )\right ) \int \frac {(d+e x)^{3/2}}{\sqrt {a+b x}} \, dx}{3 b e}\\ &=-\frac {(b B d-6 A b e+5 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{5/2}}{3 b e}-\frac {((b d-a e) (b B d-6 A b e+5 a B e)) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x}} \, dx}{8 b^2 e}\\ &=-\frac {(b d-a e) (b B d-6 A b e+5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 b^3 e}-\frac {(b B d-6 A b e+5 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{5/2}}{3 b e}-\frac {\left ((b d-a e)^2 (b B d-6 A b e+5 a B e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{16 b^3 e}\\ &=-\frac {(b d-a e) (b B d-6 A b e+5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 b^3 e}-\frac {(b B d-6 A b e+5 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{5/2}}{3 b e}-\frac {\left ((b d-a e)^2 (b B d-6 A b e+5 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 )}{8 b^4 e}\\ &=-\frac {(b d-a e) (b B d-6 A b e+5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 b^3 e}-\frac {(b B d-6 A b e+5 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{5/2}}{3 b e}-\frac {\left ((b d-a e)^2 (b B d-6 A b e+5 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 )}{8 b^4 e}\\ &=-\frac {(b d-a e) (b B d-6 A b e+5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 b^3 e}-\frac {(b B d-6 A b e+5 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{5/2}}{3 b e}-\frac {(b d-a e)^2 (b B d-6 A b e+5 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{7/2} e^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.70, size = 182, normalized size = 0.94 \[ \frac {\sqrt {d+e x} \left (\sqrt {e} \sqrt {a+b x} \left (15 a^2 B e^2-2 a b e (9 A e+11 B d+5 B e x)+b^2 \left (6 A e (5 d+2 e x)+B \left (3 d^2+14 d e x+8 e^2 x^2\right )\right )\right )-\frac {3 (b d-a e)^{3/2} (5 a B e-6 A b e+b B d) \sinh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b d-a e}}\right )}{\sqrt {\frac {b (d+e x)}{b d-a e}}}\right )}{24 b^3 e^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.75, size = 542, normalized size = 2.81 \[ \left [-\frac {3 \, {\left (B b^{3} d^{3} + 3 \, {\left (B a b^{2} - 2 \, A b^{3}\right )} d^{2} e - 3 \, {\left (3 \, B a^{2} b - 4 \, A a b^{2}\right )} d e^{2} + {\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} e^{3}\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) - 4 \, {\left (8 \, B b^{3} e^{3} x^{2} + 3 \, B b^{3} d^{2} e - 2 \, {\left (11 \, B a b^{2} - 15 \, A b^{3}\right )} d e^{2} + 3 \, {\left (5 \, B a^{2} b - 6 \, A a b^{2}\right )} e^{3} + 2 \, {\left (7 \, B b^{3} d e^{2} - {\left (5 \, B a b^{2} - 6 \, A b^{3}\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{96 \, b^{4} e^{2}}, \frac {3 \, {\left (B b^{3} d^{3} + 3 \, {\left (B a b^{2} - 2 \, A b^{3}\right )} d^{2} e - 3 \, {\left (3 \, B a^{2} b - 4 \, A a b^{2}\right )} d e^{2} + {\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} e^{3}\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) + 2 \, {\left (8 \, B b^{3} e^{3} x^{2} + 3 \, B b^{3} d^{2} e - 2 \, {\left (11 \, B a b^{2} - 15 \, A b^{3}\right )} d e^{2} + 3 \, {\left (5 \, B a^{2} b - 6 \, A a b^{2}\right )} e^{3} + 2 \, {\left (7 \, B b^{3} d e^{2} - {\left (5 \, B a b^{2} - 6 \, A b^{3}\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{48 \, b^{4} e^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.54, size = 575, normalized size = 2.98 \[ -\frac {\frac {24 \, {\left (\frac {{\left (b^{2} d - a b e\right )} e^{\left (-\frac {1}{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 )}{\sqrt {b}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a}\right )} A d {\left | b \right |}}{b^{2}} - \frac {{\left (\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{b^{2}} + \frac {{\left (b^{6} d e^{3} - 13 \, a b^{5} e^{4}\right )} e^{\left (-4\right )}}{b^{7}}\right )} - \frac {3 \, {\left (b^{7} d^{2} e^{2} + 2 \, a b^{6} d e^{3} - 11 \, a^{2} b^{5} e^{4}\right )} e^{\left (-4\right )}}{b^{7}}\right )} - \frac {3 \, {\left (b^{3} d^{3} + a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - 5 \, a^{3} e^{3}\right )} e^{\left (-\frac {5}{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 {\left | b \right |} e}{b^{2}} - \frac {6 \, {\left (\frac {{\left (b^{3} d^{2} + 2 \, a b^{2} d e - 3 \, a^{2} b e^{2}\right )} e^{\left (-\frac {3}{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 )}{\sqrt {b}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} {\left (2 \, b x + {\left (b d e - 5 \, a e^{2}\right )} e^{\left (-2\right )} + 2 \, a\right )} \sqrt {b x + a}\right )} B d {\left | b \right |}}{b^{3}} - \frac {6 \, {\left (\frac {{\left (b^{3} d^{2} + 2 \, a b^{2} d e - 3 \, a^{2} b e^{2}\right )} e^{\left (-\frac {3}{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 )}{\sqrt {b}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} {\left (2 \, b x + {\left (b d e - 5 \, a e^{2}\right )} e^{\left (-2\right )} + 2 \, a\right )} \sqrt {b x + a}\right )} A {\left | b \right |} e}{b^{3}}}{24 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 636, normalized size = 3.30 \[ \frac {\sqrt {e x +d}\, \sqrt {b x +a}\, \left (18 A \,a^{2} b \,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 )-36 A a \,b^{2} d \,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 )+18 A \,b^{3} d^{2} 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^{3} 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 )+27 B \,a^{2} b d \,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 )-9 B a \,b^{2} d^{2} 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 )-3 B \,b^{3} d^{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 )+16 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, B \,b^{2} e^{2} x^{2}+24 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, A \,b^{2} e^{2} x -20 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, B a b \,e^{2} x +28 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, B \,b^{2} d e x -36 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, A a b \,e^{2}+60 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, A \,b^{2} d e +30 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, B \,a^{2} e^{2}-44 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, B a b d e +6 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, B \,b^{2} d^{2}\right )}{48 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{3} e} \]
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 = -0.01 \[ \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{3/2}}{\sqrt {a+b\,x}} \,d x \]
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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