Optimal. Leaf size=88 \[ -\frac {2 (B d-A e)}{e \sqrt {d+e x} (b d-a e)}-\frac {2 (A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} (b d-a e)^{3/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {78, 63, 208} \[ -\frac {2 (B d-A e)}{e \sqrt {d+e x} (b d-a e)}-\frac {2 (A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} (b d-a e)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 208
Rubi steps
\begin {align*} \int \frac {A+B x}{(a+b x) (d+e x)^{3/2}} \, dx &=-\frac {2 (B d-A e)}{e (b d-a e) \sqrt {d+e x}}+\frac {(A b-a B) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{b d-a e}\\ &=-\frac {2 (B d-A e)}{e (b d-a e) \sqrt {d+e x}}+\frac {(2 (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)}\\ &=-\frac {2 (B d-A e)}{e (b d-a e) \sqrt {d+e x}}-\frac {2 (A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} (b d-a e)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 88, normalized size = 1.00 \[ \frac {2 B d-2 A e}{e \sqrt {d+e x} (a e-b d)}+\frac {2 (a B-A b) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} (b d-a e)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.02, size = 363, normalized size = 4.12 \[ \left [\frac {{\left ({\left (B a - A b\right )} e^{2} x + {\left (B a - A b\right )} d e\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e + 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) - 2 \, {\left (B b^{2} d^{2} + A a b e^{2} - {\left (B a b + A b^{2}\right )} d e\right )} \sqrt {e x + d}}{b^{3} d^{3} e - 2 \, a b^{2} d^{2} e^{2} + a^{2} b d e^{3} + {\left (b^{3} d^{2} e^{2} - 2 \, a b^{2} d e^{3} + a^{2} b e^{4}\right )} x}, -\frac {2 \, {\left ({\left ({\left (B a - A b\right )} e^{2} x + {\left (B a - A b\right )} d e\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) + {\left (B b^{2} d^{2} + A a b e^{2} - {\left (B a b + A b^{2}\right )} d e\right )} \sqrt {e x + d}\right )}}{b^{3} d^{3} e - 2 \, a b^{2} d^{2} e^{2} + a^{2} b d e^{3} + {\left (b^{3} d^{2} e^{2} - 2 \, a b^{2} d e^{3} + a^{2} b e^{4}\right )} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.25, size = 93, normalized size = 1.06 \[ -\frac {2 \, {\left (B a - A b\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} {\left (b d - a e\right )}} - \frac {2 \, {\left (B d - A e\right )}}{{\left (b d e - a e^{2}\right )} \sqrt {x e + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 142, normalized size = 1.61 \[ -\frac {2 A b \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right ) \sqrt {\left (a e -b d \right ) b}}+\frac {2 B a \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right ) \sqrt {\left (a e -b d \right ) b}}-\frac {2 A}{\left (a e -b d \right ) \sqrt {e x +d}}+\frac {2 B d}{\left (a e -b d \right ) \sqrt {e x +d}\, e} \]
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 [B] time = 0.11, size = 96, normalized size = 1.09 \[ -\frac {2\,\mathrm {atan}\left (\frac {2\,\sqrt {b}\,\left (A\,b-B\,a\right )\,\sqrt {d+e\,x}}{\left (2\,A\,b-2\,B\,a\right )\,\sqrt {a\,e-b\,d}}\right )\,\left (A\,b-B\,a\right )}{\sqrt {b}\,{\left (a\,e-b\,d\right )}^{3/2}}-\frac {2\,\left (A\,e-B\,d\right )}{e\,\left (a\,e-b\,d\right )\,\sqrt {d+e\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 30.16, size = 76, normalized size = 0.86 \[ \frac {2 \left (- A e + B d\right )}{e \sqrt {d + e x} \left (a e - b d\right )} + \frac {2 \left (- A b + B a\right ) \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e - b d}{b}}} \right )}}{b \sqrt {\frac {a e - b d}{b}} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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