Optimal. Leaf size=126 \[ -\frac {3 (5 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{7/2} \sqrt {b}}-\frac {3 (5 A b-a B)}{4 a^3 b \sqrt {x}}+\frac {5 A b-a B}{4 a^2 b \sqrt {x} (a+b x)}+\frac {A b-a B}{2 a b \sqrt {x} (a+b x)^2} \]
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Rubi [A] time = 0.05, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, 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 {3 (5 A b-a B)}{4 a^3 b \sqrt {x}}+\frac {5 A b-a B}{4 a^2 b \sqrt {x} (a+b x)}-\frac {3 (5 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{7/2} \sqrt {b}}+\frac {A b-a B}{2 a b \sqrt {x} (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}{x^{3/2} (a+b x)^3} \, dx &=\frac {A b-a B}{2 a b \sqrt {x} (a+b x)^2}-\frac {\left (-\frac {5 A b}{2}+\frac {a B}{2}\right ) \int \frac {1}{x^{3/2} (a+b x)^2} \, dx}{2 a b}\\ &=\frac {A b-a B}{2 a b \sqrt {x} (a+b x)^2}+\frac {5 A b-a B}{4 a^2 b \sqrt {x} (a+b x)}+\frac {(3 (5 A b-a B)) \int \frac {1}{x^{3/2} (a+b x)} \, dx}{8 a^2 b}\\ &=-\frac {3 (5 A b-a B)}{4 a^3 b \sqrt {x}}+\frac {A b-a B}{2 a b \sqrt {x} (a+b x)^2}+\frac {5 A b-a B}{4 a^2 b \sqrt {x} (a+b x)}-\frac {(3 (5 A b-a B)) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{8 a^3}\\ &=-\frac {3 (5 A b-a B)}{4 a^3 b \sqrt {x}}+\frac {A b-a B}{2 a b \sqrt {x} (a+b x)^2}+\frac {5 A b-a B}{4 a^2 b \sqrt {x} (a+b x)}-\frac {(3 (5 A b-a B)) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{4 a^3}\\ &=-\frac {3 (5 A b-a B)}{4 a^3 b \sqrt {x}}+\frac {A b-a B}{2 a b \sqrt {x} (a+b x)^2}+\frac {5 A b-a B}{4 a^2 b \sqrt {x} (a+b x)}-\frac {3 (5 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{7/2} \sqrt {b}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 59, normalized size = 0.47 \begin {gather*} \frac {\frac {a^2 (A b-a B)}{(a+b x)^2}+(a B-5 A b) \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};-\frac {b x}{a}\right )}{2 a^3 b \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 96, normalized size = 0.76 \begin {gather*} \frac {3 (a B-5 A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{7/2} \sqrt {b}}+\frac {-8 a^2 A+5 a^2 B x-25 a A b x+3 a b B x^2-15 A b^2 x^2}{4 a^3 \sqrt {x} (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 331, normalized size = 2.63 \begin {gather*} \left [\frac {3 \, {\left ({\left (B a b^{2} - 5 \, A b^{3}\right )} x^{3} + 2 \, {\left (B a^{2} b - 5 \, A a b^{2}\right )} x^{2} + {\left (B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a + 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) - 2 \, {\left (8 \, A a^{3} b - 3 \, {\left (B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} - 5 \, {\left (B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} \sqrt {x}}{8 \, {\left (a^{4} b^{3} x^{3} + 2 \, a^{5} b^{2} x^{2} + a^{6} b x\right )}}, -\frac {3 \, {\left ({\left (B a b^{2} - 5 \, A b^{3}\right )} x^{3} + 2 \, {\left (B a^{2} b - 5 \, A a b^{2}\right )} x^{2} + {\left (B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (8 \, A a^{3} b - 3 \, {\left (B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} - 5 \, {\left (B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} \sqrt {x}}{4 \, {\left (a^{4} b^{3} x^{3} + 2 \, a^{5} b^{2} x^{2} + a^{6} b x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.25, size = 86, normalized size = 0.68 \begin {gather*} \frac {3 \, {\left (B a - 5 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{3}} - \frac {2 \, A}{a^{3} \sqrt {x}} + \frac {3 \, B a b x^{\frac {3}{2}} - 7 \, A b^{2} x^{\frac {3}{2}} + 5 \, B a^{2} \sqrt {x} - 9 \, A a b \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 125, normalized size = 0.99 \begin {gather*} -\frac {7 A \,b^{2} x^{\frac {3}{2}}}{4 \left (b x +a \right )^{2} a^{3}}+\frac {3 B b \,x^{\frac {3}{2}}}{4 \left (b x +a \right )^{2} a^{2}}-\frac {9 A b \sqrt {x}}{4 \left (b x +a \right )^{2} a^{2}}+\frac {5 B \sqrt {x}}{4 \left (b x +a \right )^{2} a}-\frac {15 A b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}\, a^{3}}+\frac {3 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}\, a^{2}}-\frac {2 A}{a^{3} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.97, size = 98, normalized size = 0.78 \begin {gather*} -\frac {8 \, A a^{2} - 3 \, {\left (B a b - 5 \, A b^{2}\right )} x^{2} - 5 \, {\left (B a^{2} - 5 \, A a b\right )} x}{4 \, {\left (a^{3} b^{2} x^{\frac {5}{2}} + 2 \, a^{4} b x^{\frac {3}{2}} + a^{5} \sqrt {x}\right )}} + \frac {3 \, {\left (B a - 5 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.47, size = 116, normalized size = 0.92 \begin {gather*} -\frac {\frac {2\,A}{a}+\frac {5\,x\,\left (5\,A\,b-B\,a\right )}{4\,a^2}+\frac {3\,b\,x^2\,\left (5\,A\,b-B\,a\right )}{4\,a^3}}{a^2\,\sqrt {x}+b^2\,x^{5/2}+2\,a\,b\,x^{3/2}}-\frac {3\,\mathrm {atan}\left (\frac {3\,\sqrt {b}\,\sqrt {x}\,\left (5\,A\,b-B\,a\right )}{\sqrt {a}\,\left (15\,A\,b-3\,B\,a\right )}\right )\,\left (5\,A\,b-B\,a\right )}{4\,a^{7/2}\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 63.67, size = 1761, normalized size = 13.98
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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