Optimal. Leaf size=202 \[ -\frac {2 (B d-A e) (a+b x)^{5/2}}{e (b d-a e) \sqrt {d+e x}}-\frac {3 (5 b B d-4 A b e-a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 e^3}+\frac {(5 b B d-4 A b e-a B e) (a+b x)^{3/2} \sqrt {d+e x}}{2 e^2 (b d-a e)}+\frac {3 (b d-a e) (5 b B d-4 A b e-a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 \sqrt {b} e^{7/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 202, 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 = {79, 52, 65, 223,
212} \begin {gather*} \frac {3 (b d-a e) (-a B e-4 A b e+5 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 \sqrt {b} e^{7/2}}-\frac {3 \sqrt {a+b x} \sqrt {d+e x} (-a B e-4 A b e+5 b B d)}{4 e^3}+\frac {(a+b x)^{3/2} \sqrt {d+e x} (-a B e-4 A b e+5 b B d)}{2 e^2 (b d-a e)}-\frac {2 (a+b x)^{5/2} (B d-A e)}{e \sqrt {d+e x} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 79
Rule 212
Rule 223
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{3/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{e (b d-a e) \sqrt {d+e x}}+\frac {(5 b B d-4 A b e-a B e) \int \frac {(a+b x)^{3/2}}{\sqrt {d+e x}} \, dx}{e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{e (b d-a e) \sqrt {d+e x}}+\frac {(5 b B d-4 A b e-a B e) (a+b x)^{3/2} \sqrt {d+e x}}{2 e^2 (b d-a e)}-\frac {(3 (5 b B d-4 A b e-a B e)) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}} \, dx}{4 e^2}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{e (b d-a e) \sqrt {d+e x}}-\frac {3 (5 b B d-4 A b e-a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 e^3}+\frac {(5 b B d-4 A b e-a B e) (a+b x)^{3/2} \sqrt {d+e x}}{2 e^2 (b d-a e)}+\frac {(3 (b d-a e) (5 b B d-4 A b e-a B e)) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{8 e^3}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{e (b d-a e) \sqrt {d+e x}}-\frac {3 (5 b B d-4 A b e-a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 e^3}+\frac {(5 b B d-4 A b e-a B e) (a+b x)^{3/2} \sqrt {d+e x}}{2 e^2 (b d-a e)}+\frac {(3 (b d-a e) (5 b B d-4 A b e-a B e)) \text {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{4 b e^3}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{e (b d-a e) \sqrt {d+e x}}-\frac {3 (5 b B d-4 A b e-a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 e^3}+\frac {(5 b B d-4 A b e-a B e) (a+b x)^{3/2} \sqrt {d+e x}}{2 e^2 (b d-a e)}+\frac {(3 (b d-a e) (5 b B d-4 A b e-a B e)) \text {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{4 b e^3}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{e (b d-a e) \sqrt {d+e x}}-\frac {3 (5 b B d-4 A b e-a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 e^3}+\frac {(5 b B d-4 A b e-a B e) (a+b x)^{3/2} \sqrt {d+e x}}{2 e^2 (b d-a e)}+\frac {3 (b d-a e) (5 b B d-4 A b e-a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 \sqrt {b} e^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 1.18, size = 145, normalized size = 0.72 \begin {gather*} \frac {\frac {e \sqrt {a+b x} \left (4 A b e (3 d+e x)+a e (13 B d-8 A e+5 B e x)+b B \left (-15 d^2-5 d e x+2 e^2 x^2\right )\right )}{\sqrt {d+e x}}+\frac {3 (b d-a e) (-5 b B d+4 A b e+a B e) \log \left (\sqrt {a+b x}-\sqrt {\frac {b}{e}} \sqrt {d+e x}\right )}{\sqrt {\frac {b}{e}}}}{4 e^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(739\) vs.
\(2(172)=344\).
time = 0.09, size = 740, normalized size = 3.66
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \left (12 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a b \,e^{3} x -12 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{2} d \,e^{2} x +3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} e^{3} x -18 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a b d \,e^{2} x +15 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{2} d^{2} e x +4 B b \,e^{2} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+12 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a b d \,e^{2}-12 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{2} d^{2} e +8 A b \,e^{2} x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} d \,e^{2}-18 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a b \,d^{2} e +15 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{2} d^{3}+10 B a \,e^{2} x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-10 B b d e x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-16 A a \,e^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+24 A b d e \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+26 B a d e \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-30 B b \,d^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\right )}{8 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, \sqrt {e x +d}\, e^{3}}\) | \(740\) |
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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Fricas [A]
time = 2.22, size = 560, normalized size = 2.77 \begin {gather*} \left [\frac {3 \, {\left (5 \, B b^{2} d^{3} + {\left (B a^{2} + 4 \, A a b\right )} x e^{3} - {\left (2 \, {\left (3 \, B a b + 2 \, A b^{2}\right )} d x - {\left (B a^{2} + 4 \, A a b\right )} d\right )} e^{2} + {\left (5 \, B b^{2} d^{2} x - 2 \, {\left (3 \, B a b + 2 \, A b^{2}\right )} d^{2}\right )} e\right )} \sqrt {b} e^{\frac {1}{2}} \log \left (b^{2} d^{2} + 4 \, {\left (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d} \sqrt {b} e^{\frac {1}{2}} + {\left (8 \, b^{2} x^{2} + 8 \, a b x + a^{2}\right )} e^{2} + 2 \, {\left (4 \, b^{2} d x + 3 \, a b d\right )} e\right ) - 4 \, {\left (15 \, B b^{2} d^{2} e - {\left (2 \, B b^{2} x^{2} - 8 \, A a b + {\left (5 \, B a b + 4 \, A b^{2}\right )} x\right )} e^{3} + {\left (5 \, B b^{2} d x - {\left (13 \, B a b + 12 \, A b^{2}\right )} d\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}}{16 \, {\left (b x e^{5} + b d e^{4}\right )}}, -\frac {3 \, {\left (5 \, B b^{2} d^{3} + {\left (B a^{2} + 4 \, A a b\right )} x e^{3} - {\left (2 \, {\left (3 \, B a b + 2 \, A b^{2}\right )} d x - {\left (B a^{2} + 4 \, A a b\right )} d\right )} e^{2} + {\left (5 \, B b^{2} d^{2} x - 2 \, {\left (3 \, B a b + 2 \, A b^{2}\right )} d^{2}\right )} e\right )} \sqrt {-b e} \arctan \left (\frac {{\left (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {-b e} \sqrt {x e + d}}{2 \, {\left ({\left (b^{2} x^{2} + a b x\right )} e^{2} + {\left (b^{2} d x + a b d\right )} e\right )}}\right ) + 2 \, {\left (15 \, B b^{2} d^{2} e - {\left (2 \, B b^{2} x^{2} - 8 \, A a b + {\left (5 \, B a b + 4 \, A b^{2}\right )} x\right )} e^{3} + {\left (5 \, B b^{2} d x - {\left (13 \, B a b + 12 \, A b^{2}\right )} d\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}}{8 \, {\left (b x e^{5} + b d e^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \left (a + b x\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.62, size = 257, normalized size = 1.27 \begin {gather*} -\frac {3 \, {\left (5 \, B b^{2} d^{2} {\left | b \right |} - 6 \, B a b d {\left | b \right |} e - 4 \, A b^{2} d {\left | b \right |} e + B a^{2} {\left | b \right |} e^{2} + 4 \, A a b {\left | b \right |} e^{2}\right )} e^{\left (-\frac {7}{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 )}{4 \, b^{\frac {3}{2}}} + \frac {{\left ({\left (\frac {2 \, {\left (b x + a\right )} B {\left | b \right |} e^{\left (-1\right )}}{b} - \frac {{\left (5 \, B b^{2} d {\left | b \right |} e^{3} - B a b {\left | b \right |} e^{4} - 4 \, A b^{2} {\left | b \right |} e^{4}\right )} e^{\left (-5\right )}}{b^{2}}\right )} {\left (b x + a\right )} - \frac {3 \, {\left (5 \, B b^{3} d^{2} {\left | b \right |} e^{2} - 6 \, B a b^{2} d {\left | b \right |} e^{3} - 4 \, A b^{3} d {\left | b \right |} e^{3} + B a^{2} b {\left | b \right |} e^{4} + 4 \, A a b^{2} {\left | b \right |} e^{4}\right )} e^{\left (-5\right )}}{b^{2}}\right )} \sqrt {b x + a}}{4 \, \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}} \end {gather*}
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 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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