Optimal. Leaf size=119 \[ \frac {2 (A b-a B)}{\sqrt {d+e x} (b d-a e)^2}-\frac {2 (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)}-\frac {2 \sqrt {b} (A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{5/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 119, 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 {2 (A b-a B)}{\sqrt {d+e x} (b d-a e)^2}-\frac {2 (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)}-\frac {2 \sqrt {b} (A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(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) (d+e x)^{5/2}} \, dx &=-\frac {2 (B d-A e)}{3 e (b d-a e) (d+e x)^{3/2}}+\frac {(A b-a B) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{b d-a e}\\ &=-\frac {2 (B d-A e)}{3 e (b d-a e) (d+e x)^{3/2}}+\frac {2 (A b-a B)}{(b d-a e)^2 \sqrt {d+e x}}+\frac {(b (A b-a B)) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{(b d-a e)^2}\\ &=-\frac {2 (B d-A e)}{3 e (b d-a e) (d+e x)^{3/2}}+\frac {2 (A b-a B)}{(b d-a e)^2 \sqrt {d+e x}}+\frac {(2 b (A b-a B)) \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 d-A e)}{3 e (b d-a e) (d+e x)^{3/2}}+\frac {2 (A b-a B)}{(b d-a e)^2 \sqrt {d+e x}}-\frac {2 \sqrt {b} (A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 86, normalized size = 0.72 \begin {gather*} \frac {6 e (d+e x) (A b-a B) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b (d+e x)}{b d-a e}\right )-2 (b d-a e) (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.24, size = 141, normalized size = 1.18 \begin {gather*} -\frac {2 \left (A b^{3/2}-a \sqrt {b} B\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{(a e-b d)^{5/2}}-\frac {2 \left (a A e^2+3 a B e (d+e x)-a B d e-3 A b e (d+e x)-A b d e+b B d^2\right )}{3 e (d+e x)^{3/2} (a e-b d)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.43, size = 506, normalized size = 4.25 \begin {gather*} \left [-\frac {3 \, {\left ({\left (B a - A b\right )} e^{3} x^{2} + 2 \, {\left (B a - A b\right )} d e^{2} x + {\left (B a - A b\right )} d^{2} e\right )} \sqrt {\frac {b}{b d - a e}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, {\left (b d - a e\right )} \sqrt {e x + d} \sqrt {\frac {b}{b d - a e}}}{b x + a}\right ) + 2 \, {\left (B b d^{2} + A a e^{2} + 3 \, {\left (B a - A b\right )} e^{2} x + 2 \, {\left (B a - 2 \, A b\right )} d e\right )} \sqrt {e x + d}}{3 \, {\left (b^{2} d^{4} e - 2 \, a b d^{3} e^{2} + a^{2} d^{2} e^{3} + {\left (b^{2} d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}\right )} x^{2} + 2 \, {\left (b^{2} d^{3} e^{2} - 2 \, a b d^{2} e^{3} + a^{2} d e^{4}\right )} x\right )}}, \frac {2 \, {\left (3 \, {\left ({\left (B a - A b\right )} e^{3} x^{2} + 2 \, {\left (B a - A b\right )} d e^{2} x + {\left (B a - A b\right )} d^{2} e\right )} \sqrt {-\frac {b}{b d - a e}} \arctan \left (-\frac {{\left (b d - a e\right )} \sqrt {e x + d} \sqrt {-\frac {b}{b d - a e}}}{b e x + b d}\right ) - {\left (B b d^{2} + A a e^{2} + 3 \, {\left (B a - A b\right )} e^{2} x + 2 \, {\left (B a - 2 \, A b\right )} d e\right )} \sqrt {e x + d}\right )}}{3 \, {\left (b^{2} d^{4} e - 2 \, a b d^{3} e^{2} + a^{2} d^{2} e^{3} + {\left (b^{2} d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}\right )} x^{2} + 2 \, {\left (b^{2} d^{3} e^{2} - 2 \, a b d^{2} e^{3} + a^{2} d e^{4}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.30, size = 161, normalized size = 1.35 \begin {gather*} -\frac {2 \, {\left (B a b - A b^{2}\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 (B b d^{2} + 3 \, {\left (x e + d\right )} B a e - 3 \, {\left (x e + d\right )} A b e - B a d e - A b d e + A a e^{2}\right )}}{3 \, {\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 187, normalized size = 1.57 \begin {gather*} \frac {2 A \,b^{2} \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 a b \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 A b}{\left (a e -b d \right )^{2} \sqrt {e x +d}}-\frac {2 B a}{\left (a e -b d \right )^{2} \sqrt {e x +d}}-\frac {2 A}{3 \left (a e -b d \right ) \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 B d}{3 \left (a e -b d \right ) \left (e x +d \right )^{\frac {3}{2}} e} \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 = 1.31, size = 128, normalized size = 1.08 \begin {gather*} \frac {2\,\sqrt {b}\,\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 (A\,b-B\,a\right )}{{\left (a\,e-b\,d\right )}^{5/2}}-\frac {\frac {2\,\left (A\,e-B\,d\right )}{3\,\left (a\,e-b\,d\right )}-\frac {2\,\left (A\,b\,e-B\,a\,e\right )\,\left (d+e\,x\right )}{{\left (a\,e-b\,d\right )}^2}}{e\,{\left (d+e\,x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 43.57, size = 105, normalized size = 0.88 \begin {gather*} - \frac {2 \left (- A b + B a\right )}{\sqrt {d + e x} \left (a e - b d\right )^{2}} - \frac {2 \left (- A b + B a\right ) \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e - b d}{b}}} \right )}}{\sqrt {\frac {a e - b d}{b}} \left (a e - b d\right )^{2}} + \frac {2 \left (- A e + B d\right )}{3 e \left (d + e x\right )^{\frac {3}{2}} \left (a e - b d\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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