Optimal. Leaf size=98 \[ -\frac {2 (A b-a B) \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{5/2}}+\frac {2 \sqrt {d+e x} (A b-a B)}{b^2}+\frac {2 B (d+e x)^{3/2}}{3 b e} \]
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Rubi [A] time = 0.06, antiderivative size = 98, 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 = {80, 50, 63, 208} \[ \frac {2 \sqrt {d+e x} (A b-a B)}{b^2}-\frac {2 (A b-a B) \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{5/2}}+\frac {2 B (d+e x)^{3/2}}{3 b e} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 208
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {d+e x}}{a+b x} \, dx &=\frac {2 B (d+e x)^{3/2}}{3 b e}+\frac {\left (2 \left (\frac {3 A b e}{2}-\frac {3 a B e}{2}\right )\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{3 b e}\\ &=\frac {2 (A b-a B) \sqrt {d+e x}}{b^2}+\frac {2 B (d+e x)^{3/2}}{3 b e}+\frac {((A b-a B) (b d-a e)) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{b^2}\\ &=\frac {2 (A b-a B) \sqrt {d+e x}}{b^2}+\frac {2 B (d+e x)^{3/2}}{3 b e}+\frac {(2 (A b-a B) (b d-a 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 (A b-a B) \sqrt {d+e x}}{b^2}+\frac {2 B (d+e x)^{3/2}}{3 b e}-\frac {2 (A b-a B) \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 94, normalized size = 0.96 \[ \frac {2 (a B-A b) \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{5/2}}+\frac {2 \sqrt {d+e x} (-3 a B e+3 A b e+b B (d+e x))}{3 b^2 e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 211, normalized size = 2.15 \[ \left [-\frac {3 \, {\left (B a - A b\right )} e \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) - 2 \, {\left (B b e x + B b d - 3 \, {\left (B a - A b\right )} e\right )} \sqrt {e x + d}}{3 \, b^{2} e}, \frac {2 \, {\left (3 \, {\left (B a - A b\right )} e \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) + {\left (B b e x + B b d - 3 \, {\left (B a - A b\right )} e\right )} \sqrt {e x + d}\right )}}{3 \, b^{2} e}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.23, size = 126, normalized size = 1.29 \[ -\frac {2 \, {\left (B a b d - A b^{2} d - B a^{2} e + A a b e\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{2}} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} B b^{2} e^{2} - 3 \, \sqrt {x e + d} B a b e^{3} + 3 \, \sqrt {x e + d} A b^{2} e^{3}\right )} e^{\left (-3\right )}}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 211, normalized size = 2.15 \[ -\frac {2 A a e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b}+\frac {2 A d \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}}+\frac {2 B \,a^{2} e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{2}}-\frac {2 B a d \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b}+\frac {2 \sqrt {e x +d}\, A}{b}-\frac {2 \sqrt {e x +d}\, B a}{b^{2}}+\frac {2 \left (e x +d \right )^{\frac {3}{2}} B}{3 b e} \]
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 = 1.30, size = 107, normalized size = 1.09 \[ \left (\frac {2\,A\,e-2\,B\,d}{b\,e}-\frac {2\,B\,\left (a\,e^2-b\,d\,e\right )}{b^2\,e^2}\right )\,\sqrt {d+e\,x}-\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {b\,d-a\,e}}\right )\,\left (A\,b-B\,a\right )\,\sqrt {b\,d-a\,e}}{b^{5/2}}+\frac {2\,B\,{\left (d+e\,x\right )}^{3/2}}{3\,b\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.07, size = 94, normalized size = 0.96 \[ \frac {2 \left (\frac {B \left (d + e x\right )^{\frac {3}{2}}}{3 b} + \frac {\sqrt {d + e x} \left (A b e - B a e\right )}{b^{2}} + \frac {e \left (- A b + B a\right ) \left (a e - b d\right ) \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e - b d}{b}}} \right )}}{b^{3} \sqrt {\frac {a e - b d}{b}}}\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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