Optimal. Leaf size=100 \[ \frac {(3 a B+A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{3/2} b^{5/2}}-\frac {\sqrt {x} (3 a B+A b)}{4 a b^2 (a+b x)}+\frac {x^{3/2} (A b-a B)}{2 a b (a+b x)^2} \]
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Rubi [A] time = 0.04, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {78, 47, 63, 205} \begin {gather*} \frac {(3 a B+A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{3/2} b^{5/2}}-\frac {\sqrt {x} (3 a B+A b)}{4 a b^2 (a+b x)}+\frac {x^{3/2} (A b-a B)}{2 a b (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 78
Rule 205
Rubi steps
\begin {align*} \int \frac {\sqrt {x} (A+B x)}{(a+b x)^3} \, dx &=\frac {(A b-a B) x^{3/2}}{2 a b (a+b x)^2}+\frac {(A b+3 a B) \int \frac {\sqrt {x}}{(a+b x)^2} \, dx}{4 a b}\\ &=\frac {(A b-a B) x^{3/2}}{2 a b (a+b x)^2}-\frac {(A b+3 a B) \sqrt {x}}{4 a b^2 (a+b x)}+\frac {(A b+3 a B) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{8 a b^2}\\ &=\frac {(A b-a B) x^{3/2}}{2 a b (a+b x)^2}-\frac {(A b+3 a B) \sqrt {x}}{4 a b^2 (a+b x)}+\frac {(A b+3 a B) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{4 a b^2}\\ &=\frac {(A b-a B) x^{3/2}}{2 a b (a+b x)^2}-\frac {(A b+3 a B) \sqrt {x}}{4 a b^2 (a+b x)}+\frac {(A b+3 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{3/2} b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 85, normalized size = 0.85 \begin {gather*} \frac {(3 a B+A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{3/2} b^{5/2}}+\frac {\sqrt {x} \left (-3 a^2 B-a b (A+5 B x)+A b^2 x\right )}{4 a b^2 (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 86, normalized size = 0.86 \begin {gather*} \frac {(3 a B+A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{3/2} b^{5/2}}-\frac {\sqrt {x} \left (3 a^2 B+a A b+5 a b B x-A b^2 x\right )}{4 a b^2 (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 291, normalized size = 2.91 \begin {gather*} \left [-\frac {{\left (3 \, B a^{3} + A a^{2} b + {\left (3 \, B a b^{2} + A b^{3}\right )} x^{2} + 2 \, {\left (3 \, B a^{2} b + A a b^{2}\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) + 2 \, {\left (3 \, B a^{3} b + A a^{2} b^{2} + {\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} x\right )} \sqrt {x}}{8 \, {\left (a^{2} b^{5} x^{2} + 2 \, a^{3} b^{4} x + a^{4} b^{3}\right )}}, -\frac {{\left (3 \, B a^{3} + A a^{2} b + {\left (3 \, B a b^{2} + A b^{3}\right )} x^{2} + 2 \, {\left (3 \, B a^{2} b + A a b^{2}\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (3 \, B a^{3} b + A a^{2} b^{2} + {\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} x\right )} \sqrt {x}}{4 \, {\left (a^{2} b^{5} x^{2} + 2 \, a^{3} b^{4} x + a^{4} b^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.24, size = 82, normalized size = 0.82 \begin {gather*} \frac {{\left (3 \, B a + A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a b^{2}} - \frac {5 \, B a b x^{\frac {3}{2}} - A b^{2} x^{\frac {3}{2}} + 3 \, B a^{2} \sqrt {x} + A a b \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} a b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 94, normalized size = 0.94 \begin {gather*} \frac {A \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}\, a b}+\frac {3 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}\, b^{2}}+\frac {\frac {\left (A b -5 B a \right ) x^{\frac {3}{2}}}{4 a b}-\frac {\left (A b +3 B a \right ) \sqrt {x}}{4 b^{2}}}{\left (b x +a \right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.99, size = 94, normalized size = 0.94 \begin {gather*} -\frac {{\left (5 \, B a b - A b^{2}\right )} x^{\frac {3}{2}} + {\left (3 \, B a^{2} + A a b\right )} \sqrt {x}}{4 \, {\left (a b^{4} x^{2} + 2 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} + \frac {{\left (3 \, B a + A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 84, normalized size = 0.84 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (A\,b+3\,B\,a\right )}{4\,a^{3/2}\,b^{5/2}}-\frac {\frac {\sqrt {x}\,\left (A\,b+3\,B\,a\right )}{4\,b^2}-\frac {x^{3/2}\,\left (A\,b-5\,B\,a\right )}{4\,a\,b}}{a^2+2\,a\,b\,x+b^2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 20.73, size = 1499, normalized size = 14.99
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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