Optimal. Leaf size=103 \[ -\frac {(-a B e-A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} (b d-a e)^{3/2}}-\frac {\sqrt {d+e x} (A b-a B)}{b (a+b x) (b d-a e)} \]
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Rubi [A] time = 0.10, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {27, 78, 63, 208} \[ -\frac {(-a B e-A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} (b d-a e)^{3/2}}-\frac {\sqrt {d+e x} (A b-a B)}{b (a+b x) (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 63
Rule 78
Rule 208
Rubi steps
\begin {align*} \int \frac {A+B x}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac {A+B x}{(a+b x)^2 \sqrt {d+e x}} \, dx\\ &=-\frac {(A b-a B) \sqrt {d+e x}}{b (b d-a e) (a+b x)}+\frac {(2 b B d-A b e-a B e) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{2 b (b d-a e)}\\ &=-\frac {(A b-a B) \sqrt {d+e x}}{b (b d-a e) (a+b x)}+\frac {(2 b B d-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 )}{b e (b d-a e)}\\ &=-\frac {(A b-a B) \sqrt {d+e x}}{b (b d-a e) (a+b x)}-\frac {(2 b B d-A b e-a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} (b d-a e)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 102, normalized size = 0.99 \[ \frac {\sqrt {d+e x} (a B-A b)}{b (a+b x) (b d-a e)}-\frac {(-a B e-A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} (b d-a e)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.99, size = 397, normalized size = 3.85 \[ \left [\frac {{\left (2 \, B a b d - {\left (B a^{2} + A a b\right )} e + {\left (2 \, B b^{2} d - {\left (B a b + A b^{2}\right )} e\right )} x\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) + 2 \, {\left ({\left (B a b^{2} - A b^{3}\right )} d - {\left (B a^{2} b - A a b^{2}\right )} e\right )} \sqrt {e x + d}}{2 \, {\left (a b^{4} d^{2} - 2 \, a^{2} b^{3} d e + a^{3} b^{2} e^{2} + {\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} x\right )}}, \frac {{\left (2 \, B a b d - {\left (B a^{2} + A a b\right )} e + {\left (2 \, B b^{2} d - {\left (B a b + A b^{2}\right )} e\right )} x\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) + {\left ({\left (B a b^{2} - A b^{3}\right )} d - {\left (B a^{2} b - A a b^{2}\right )} e\right )} \sqrt {e x + d}}{a b^{4} d^{2} - 2 \, a^{2} b^{3} d e + a^{3} b^{2} e^{2} + {\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 135, normalized size = 1.31 \[ \frac {{\left (2 \, B b d - B a e - A b e\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{{\left (b^{2} d - a b e\right )} \sqrt {-b^{2} d + a b e}} + \frac {\sqrt {x e + d} B a e - \sqrt {x e + d} A b e}{{\left (b^{2} d - a b e\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 195, normalized size = 1.89 \[ \frac {A e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right ) \sqrt {\left (a e -b d \right ) b}}+\frac {B a e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right ) \sqrt {\left (a e -b d \right ) b}\, b}-\frac {2 B d \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right ) \sqrt {\left (a e -b d \right ) b}}+\frac {\left (A b -B a \right ) \sqrt {e x +d}\, e}{\left (a e -b d \right ) \left (a e -b d +\left (e x +d \right ) b \right ) b} \]
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 = 0.15, size = 99, normalized size = 0.96 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )\,\left (A\,b\,e+B\,a\,e-2\,B\,b\,d\right )}{b^{3/2}\,{\left (a\,e-b\,d\right )}^{3/2}}+\frac {\left (A\,b\,e-B\,a\,e\right )\,\sqrt {d+e\,x}}{b\,\left (a\,e-b\,d\right )\,\left (a\,e-b\,d+b\,\left (d+e\,x\right )\right )} \]
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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