Optimal. Leaf size=140 \[ -\frac {(-3 a B e+A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{5/2} \sqrt {b d-a e}}+\frac {\sqrt {d+e x} (-3 a B e+A b e+2 b B d)}{b^2 (b d-a e)}-\frac {(d+e x)^{3/2} (A b-a B)}{b (a+b x) (b d-a e)} \]
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Rubi [A] time = 0.11, 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, 50, 63, 208} \[ \frac {\sqrt {d+e x} (-3 a B e+A b e+2 b B d)}{b^2 (b d-a e)}-\frac {(-3 a B e+A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{5/2} \sqrt {b d-a e}}-\frac {(d+e x)^{3/2} (A b-a B)}{b (a+b x) (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
Rule 208
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^2} \, dx &=-\frac {(A b-a B) (d+e x)^{3/2}}{b (b d-a e) (a+b x)}+\frac {(2 b B d+A b e-3 a B e) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{2 b (b d-a e)}\\ &=\frac {(2 b B d+A b e-3 a B e) \sqrt {d+e x}}{b^2 (b d-a e)}-\frac {(A b-a B) (d+e x)^{3/2}}{b (b d-a e) (a+b x)}+\frac {(2 b B d+A b e-3 a B e) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{2 b^2}\\ &=\frac {(2 b B d+A b e-3 a B e) \sqrt {d+e x}}{b^2 (b d-a e)}-\frac {(A b-a B) (d+e x)^{3/2}}{b (b d-a e) (a+b x)}+\frac {(2 b B d+A b e-3 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^2 e}\\ &=\frac {(2 b B d+A b e-3 a B e) \sqrt {d+e x}}{b^2 (b d-a e)}-\frac {(A b-a B) (d+e x)^{3/2}}{b (b d-a e) (a+b x)}-\frac {(2 b B d+A b e-3 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{5/2} \sqrt {b d-a e}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 119, normalized size = 0.85 \[ \frac {\frac {(-3 a B e+A b e+2 b B d) \left (\sqrt {b} \sqrt {d+e x}-\sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )\right )}{b^{3/2}}+\frac {(d+e x)^{3/2} (a B-A b)}{a+b x}}{b (b d-a e)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 393, normalized size = 2.81 \[ \left [\frac {{\left (2 \, B a b d - {\left (3 \, B a^{2} - A a b\right )} e + {\left (2 \, B b^{2} d - {\left (3 \, B a b - A b^{2}\right )} e\right )} x\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) + 2 \, {\left ({\left (3 \, B a b^{2} - A b^{3}\right )} d - {\left (3 \, B a^{2} b - A a b^{2}\right )} e + 2 \, {\left (B b^{3} d - B a b^{2} e\right )} x\right )} \sqrt {e x + d}}{2 \, {\left (a b^{4} d - a^{2} b^{3} e + {\left (b^{5} d - a b^{4} e\right )} x\right )}}, \frac {{\left (2 \, B a b d - {\left (3 \, B a^{2} - A a b\right )} e + {\left (2 \, B b^{2} d - {\left (3 \, B a b - A b^{2}\right )} e\right )} x\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) + {\left ({\left (3 \, B a b^{2} - A b^{3}\right )} d - {\left (3 \, B a^{2} b - A a b^{2}\right )} e + 2 \, {\left (B b^{3} d - B a b^{2} e\right )} x\right )} \sqrt {e x + d}}{a b^{4} d - a^{2} b^{3} e + {\left (b^{5} d - a b^{4} e\right )} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.30, size = 126, normalized size = 0.90 \[ \frac {2 \, \sqrt {x e + d} B}{b^{2}} + \frac {{\left (2 \, B b d - 3 \, B a e + A b e\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{2}} + \frac {\sqrt {x e + d} B a e - \sqrt {x e + d} A b e}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 186, normalized size = 1.33 \[ \frac {A e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b}-\frac {3 B a e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{2}}+\frac {2 B d \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b}-\frac {\sqrt {e x +d}\, A e}{\left (b x e +a e \right ) b}+\frac {\sqrt {e x +d}\, B a e}{\left (b x e +a e \right ) b^{2}}+\frac {2 \sqrt {e x +d}\, B}{b^{2}} \]
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 = 0.14, size = 108, normalized size = 0.77 \[ \frac {2\,B\,\sqrt {d+e\,x}}{b^2}-\frac {\left (A\,b\,e-B\,a\,e\right )\,\sqrt {d+e\,x}}{b^3\,\left (d+e\,x\right )-b^3\,d+a\,b^2\,e}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )\,\left (A\,b\,e-3\,B\,a\,e+2\,B\,b\,d\right )}{b^{5/2}\,\sqrt {a\,e-b\,d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 111.76, size = 1251, normalized size = 8.94 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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