Optimal. Leaf size=100 \[ \frac {(a B+3 A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} b^{3/2}}+\frac {\sqrt {x} (a B+3 A b)}{4 a^2 b (a+b x)}+\frac {\sqrt {x} (A b-a B)}{2 a b (a+b x)^2} \]
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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, 51, 63, 205} \begin {gather*} \frac {(a B+3 A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} b^{3/2}}+\frac {\sqrt {x} (a B+3 A b)}{4 a^2 b (a+b x)}+\frac {\sqrt {x} (A b-a B)}{2 a b (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 205
Rubi steps
\begin {align*} \int \frac {A+B x}{\sqrt {x} (a+b x)^3} \, dx &=\frac {(A b-a B) \sqrt {x}}{2 a b (a+b x)^2}+\frac {(3 A b+a B) \int \frac {1}{\sqrt {x} (a+b x)^2} \, dx}{4 a b}\\ &=\frac {(A b-a B) \sqrt {x}}{2 a b (a+b x)^2}+\frac {(3 A b+a B) \sqrt {x}}{4 a^2 b (a+b x)}+\frac {(3 A b+a B) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{8 a^2 b}\\ &=\frac {(A b-a B) \sqrt {x}}{2 a b (a+b x)^2}+\frac {(3 A b+a B) \sqrt {x}}{4 a^2 b (a+b x)}+\frac {(3 A b+a B) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{4 a^2 b}\\ &=\frac {(A b-a B) \sqrt {x}}{2 a b (a+b x)^2}+\frac {(3 A b+a B) \sqrt {x}}{4 a^2 b (a+b x)}+\frac {(3 A b+a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 91, normalized size = 0.91 \begin {gather*} \frac {\sqrt {x} \left (\frac {a^2 (A b-a B)}{(a+b x)^2}-\frac {1}{2} (-a B-3 A b) \left (\frac {a}{a+b x}+\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {b} \sqrt {x}}\right )\right )}{2 a^3 b} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 86, normalized size = 0.86 \begin {gather*} \frac {(a B+3 A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} b^{3/2}}-\frac {\sqrt {x} \left (a^2 B-5 a A b-a b B x-3 A b^2 x\right )}{4 a^2 b (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.13, size = 291, normalized size = 2.91 \begin {gather*} \left [-\frac {{\left (B a^{3} + 3 \, A a^{2} b + {\left (B a b^{2} + 3 \, A b^{3}\right )} x^{2} + 2 \, {\left (B a^{2} b + 3 \, 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 (B a^{3} b - 5 \, A a^{2} b^{2} - {\left (B a^{2} b^{2} + 3 \, A a b^{3}\right )} x\right )} \sqrt {x}}{8 \, {\left (a^{3} b^{4} x^{2} + 2 \, a^{4} b^{3} x + a^{5} b^{2}\right )}}, -\frac {{\left (B a^{3} + 3 \, A a^{2} b + {\left (B a b^{2} + 3 \, A b^{3}\right )} x^{2} + 2 \, {\left (B a^{2} b + 3 \, A a b^{2}\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (B a^{3} b - 5 \, A a^{2} b^{2} - {\left (B a^{2} b^{2} + 3 \, A a b^{3}\right )} x\right )} \sqrt {x}}{4 \, {\left (a^{3} b^{4} x^{2} + 2 \, a^{4} b^{3} x + a^{5} b^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.29, size = 82, normalized size = 0.82 \begin {gather*} \frac {{\left (B a + 3 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{2} b} + \frac {B a b x^{\frac {3}{2}} + 3 \, A b^{2} x^{\frac {3}{2}} - B a^{2} \sqrt {x} + 5 \, A a b \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} a^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 95, normalized size = 0.95 \begin {gather*} \frac {3 A \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}\, a^{2}}+\frac {B \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}\, a b}+\frac {\frac {\left (3 A b +B a \right ) x^{\frac {3}{2}}}{4 a^{2}}+\frac {\left (5 A b -B a \right ) \sqrt {x}}{4 a b}}{\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 = 2.03, size = 94, normalized size = 0.94 \begin {gather*} \frac {{\left (B a b + 3 \, A b^{2}\right )} x^{\frac {3}{2}} - {\left (B a^{2} - 5 \, A a b\right )} \sqrt {x}}{4 \, {\left (a^{2} b^{3} x^{2} + 2 \, a^{3} b^{2} x + a^{4} b\right )}} + \frac {{\left (B a + 3 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 84, normalized size = 0.84 \begin {gather*} \frac {\frac {x^{3/2}\,\left (3\,A\,b+B\,a\right )}{4\,a^2}+\frac {\sqrt {x}\,\left (5\,A\,b-B\,a\right )}{4\,a\,b}}{a^2+2\,a\,b\,x+b^2\,x^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (3\,A\,b+B\,a\right )}{4\,a^{5/2}\,b^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 28.98, size = 1501, normalized size = 15.01
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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