Optimal. Leaf size=140 \[ -\frac {A b-a B}{b (a+b x) \sqrt {d+e x} (b d-a e)}+\frac {a B e-3 A b e+2 b B d}{b \sqrt {d+e x} (b d-a e)^2}-\frac {(a B e-3 A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} (b d-a e)^{5/2}} \]
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Rubi [A] time = 0.13, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {78, 51, 63, 208} \begin {gather*} -\frac {A b-a B}{b (a+b x) \sqrt {d+e x} (b d-a e)}+\frac {a B e-3 A b e+2 b B d}{b \sqrt {d+e x} (b d-a e)^2}-\frac {(a B e-3 A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} (b d-a e)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 208
Rubi steps
\begin {align*} \int \frac {A+B x}{(a+b x)^2 (d+e x)^{3/2}} \, dx &=-\frac {A b-a B}{b (b d-a e) (a+b x) \sqrt {d+e x}}+\frac {(2 b B d-3 A b e+a B e) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{2 b (b d-a e)}\\ &=\frac {2 b B d-3 A b e+a B e}{b (b d-a e)^2 \sqrt {d+e x}}-\frac {A b-a B}{b (b d-a e) (a+b x) \sqrt {d+e x}}+\frac {(2 b B d-3 A b e+a B e) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{2 (b d-a e)^2}\\ &=\frac {2 b B d-3 A b e+a B e}{b (b d-a e)^2 \sqrt {d+e x}}-\frac {A b-a B}{b (b d-a e) (a+b x) \sqrt {d+e x}}+\frac {(2 b B d-3 A b e+a B e) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{e (b d-a e)^2}\\ &=\frac {2 b B d-3 A b e+a B e}{b (b d-a e)^2 \sqrt {d+e x}}-\frac {A b-a B}{b (b d-a e) (a+b x) \sqrt {d+e x}}-\frac {(2 b B d-3 A b e+a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} (b d-a e)^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 95, normalized size = 0.68 \begin {gather*} \frac {(a+b x) (a B e-3 A b e+2 b B d) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b (d+e x)}{b d-a e}\right )-(A b-a B) (b d-a e)}{b (a+b x) \sqrt {d+e x} (b d-a e)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.59, size = 178, normalized size = 1.27 \begin {gather*} \frac {2 a A e^2-a B e (d+e x)-2 a B d e+3 A b e (d+e x)-2 A b d e+2 b B d^2-2 b B d (d+e x)}{\sqrt {d+e x} (b d-a e)^2 (-a e-b (d+e x)+b d)}+\frac {(-a B e+3 A b e-2 b B d) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{\sqrt {b} (b d-a e)^2 \sqrt {a e-b d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.67, size = 775, normalized size = 5.54 \begin {gather*} \left [-\frac {{\left (2 \, B a b d^{2} + {\left (B a^{2} - 3 \, A a b\right )} d e + {\left (2 \, B b^{2} d e + {\left (B a b - 3 \, A b^{2}\right )} e^{2}\right )} x^{2} + {\left (2 \, B b^{2} d^{2} + 3 \, {\left (B a b - A b^{2}\right )} d e + {\left (B a^{2} - 3 \, A a b\right )} e^{2}\right )} x\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e + 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) - 2 \, {\left (2 \, A a^{2} b e^{2} + {\left (3 \, B a b^{2} - A b^{3}\right )} d^{2} - {\left (3 \, B a^{2} b + A a b^{2}\right )} d e + {\left (2 \, B b^{3} d^{2} - {\left (B a b^{2} + 3 \, A b^{3}\right )} d e - {\left (B a^{2} b - 3 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{2 \, {\left (a b^{4} d^{4} - 3 \, a^{2} b^{3} d^{3} e + 3 \, a^{3} b^{2} d^{2} e^{2} - a^{4} b d e^{3} + {\left (b^{5} d^{3} e - 3 \, a b^{4} d^{2} e^{2} + 3 \, a^{2} b^{3} d e^{3} - a^{3} b^{2} e^{4}\right )} x^{2} + {\left (b^{5} d^{4} - 2 \, a b^{4} d^{3} e + 2 \, a^{3} b^{2} d e^{3} - a^{4} b e^{4}\right )} x\right )}}, \frac {{\left (2 \, B a b d^{2} + {\left (B a^{2} - 3 \, A a b\right )} d e + {\left (2 \, B b^{2} d e + {\left (B a b - 3 \, A b^{2}\right )} e^{2}\right )} x^{2} + {\left (2 \, B b^{2} d^{2} + 3 \, {\left (B a b - A b^{2}\right )} d e + {\left (B a^{2} - 3 \, A a b\right )} e^{2}\right )} x\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) + {\left (2 \, A a^{2} b e^{2} + {\left (3 \, B a b^{2} - A b^{3}\right )} d^{2} - {\left (3 \, B a^{2} b + A a b^{2}\right )} d e + {\left (2 \, B b^{3} d^{2} - {\left (B a b^{2} + 3 \, A b^{3}\right )} d e - {\left (B a^{2} b - 3 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{a b^{4} d^{4} - 3 \, a^{2} b^{3} d^{3} e + 3 \, a^{3} b^{2} d^{2} e^{2} - a^{4} b d e^{3} + {\left (b^{5} d^{3} e - 3 \, a b^{4} d^{2} e^{2} + 3 \, a^{2} b^{3} d e^{3} - a^{3} b^{2} e^{4}\right )} x^{2} + {\left (b^{5} d^{4} - 2 \, a b^{4} d^{3} e + 2 \, a^{3} b^{2} d e^{3} - a^{4} b e^{4}\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.30, size = 204, normalized size = 1.46 \begin {gather*} \frac {{\left (2 \, B b d + B a e - 3 \, A b e\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \sqrt {-b^{2} d + a b e}} + \frac {2 \, {\left (x e + d\right )} B b d - 2 \, B b d^{2} + {\left (x e + d\right )} B a e - 3 \, {\left (x e + d\right )} A b e + 2 \, B a d e + 2 \, A b d e - 2 \, A a e^{2}}{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} {\left ({\left (x e + d\right )}^{\frac {3}{2}} b - \sqrt {x e + d} b d + \sqrt {x e + d} a e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 253, normalized size = 1.81 \begin {gather*} -\frac {3 A b e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{2} \sqrt {\left (a e -b d \right ) b}}+\frac {B a e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{2} \sqrt {\left (a e -b d \right ) b}}+\frac {2 B b d \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{2} \sqrt {\left (a e -b d \right ) b}}-\frac {\sqrt {e x +d}\, A b e}{\left (a e -b d \right )^{2} \left (b x e +a e \right )}+\frac {\sqrt {e x +d}\, B a e}{\left (a e -b d \right )^{2} \left (b x e +a e \right )}-\frac {2 A e}{\left (a e -b d \right )^{2} \sqrt {e x +d}}+\frac {2 B d}{\left (a e -b d \right )^{2} \sqrt {e x +d}} \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 = 0.20, size = 156, normalized size = 1.11 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}{{\left (a\,e-b\,d\right )}^{5/2}}\right )\,\left (B\,a\,e-3\,A\,b\,e+2\,B\,b\,d\right )}{\sqrt {b}\,{\left (a\,e-b\,d\right )}^{5/2}}-\frac {\frac {2\,\left (A\,e-B\,d\right )}{a\,e-b\,d}-\frac {\left (d+e\,x\right )\,\left (B\,a\,e-3\,A\,b\,e+2\,B\,b\,d\right )}{{\left (a\,e-b\,d\right )}^2}}{b\,{\left (d+e\,x\right )}^{3/2}+\left (a\,e-b\,d\right )\,\sqrt {d+e\,x}} \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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