Optimal. Leaf size=110 \[ -\frac {3 b (5 A b-4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 a^{7/2}}+\frac {3 b (5 A b-4 a B)}{4 a^3 \sqrt {a+b x}}+\frac {5 A b-4 a B}{4 a^2 x \sqrt {a+b x}}-\frac {A}{2 a x^2 \sqrt {a+b x}} \]
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Rubi [A] time = 0.04, antiderivative size = 112, normalized size of antiderivative = 1.02, 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, 208} \begin {gather*} \frac {3 \sqrt {a+b x} (5 A b-4 a B)}{4 a^3 x}-\frac {5 A b-4 a B}{2 a^2 x \sqrt {a+b x}}-\frac {3 b (5 A b-4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 a^{7/2}}-\frac {A}{2 a x^2 \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 208
Rubi steps
\begin {align*} \int \frac {A+B x}{x^3 (a+b x)^{3/2}} \, dx &=-\frac {A}{2 a x^2 \sqrt {a+b x}}+\frac {\left (-\frac {5 A b}{2}+2 a B\right ) \int \frac {1}{x^2 (a+b x)^{3/2}} \, dx}{2 a}\\ &=-\frac {A}{2 a x^2 \sqrt {a+b x}}-\frac {5 A b-4 a B}{2 a^2 x \sqrt {a+b x}}-\frac {(3 (5 A b-4 a B)) \int \frac {1}{x^2 \sqrt {a+b x}} \, dx}{4 a^2}\\ &=-\frac {A}{2 a x^2 \sqrt {a+b x}}-\frac {5 A b-4 a B}{2 a^2 x \sqrt {a+b x}}+\frac {3 (5 A b-4 a B) \sqrt {a+b x}}{4 a^3 x}+\frac {(3 b (5 A b-4 a B)) \int \frac {1}{x \sqrt {a+b x}} \, dx}{8 a^3}\\ &=-\frac {A}{2 a x^2 \sqrt {a+b x}}-\frac {5 A b-4 a B}{2 a^2 x \sqrt {a+b x}}+\frac {3 (5 A b-4 a B) \sqrt {a+b x}}{4 a^3 x}+\frac {(3 (5 A b-4 a B)) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{4 a^3}\\ &=-\frac {A}{2 a x^2 \sqrt {a+b x}}-\frac {5 A b-4 a B}{2 a^2 x \sqrt {a+b x}}+\frac {3 (5 A b-4 a B) \sqrt {a+b x}}{4 a^3 x}-\frac {3 b (5 A b-4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 a^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 56, normalized size = 0.51 \begin {gather*} \frac {b x^2 (5 A b-4 a B) \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};\frac {b x}{a}+1\right )-a^2 A}{2 a^3 x^2 \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 117, normalized size = 1.06 \begin {gather*} \frac {3 \left (4 a b B-5 A b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 a^{7/2}}-\frac {8 a^3 B-8 a^2 A b-20 a^2 B (a+b x)+25 a A b (a+b x)-15 A b (a+b x)^2+12 a B (a+b x)^2}{4 a^3 b x^2 \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.41, size = 275, normalized size = 2.50 \begin {gather*} \left [-\frac {3 \, {\left ({\left (4 \, B a b^{2} - 5 \, A b^{3}\right )} x^{3} + {\left (4 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{2}\right )} \sqrt {a} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (2 \, A a^{3} + 3 \, {\left (4 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{2} + {\left (4 \, B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt {b x + a}}{8 \, {\left (a^{4} b x^{3} + a^{5} x^{2}\right )}}, -\frac {3 \, {\left ({\left (4 \, B a b^{2} - 5 \, A b^{3}\right )} x^{3} + {\left (4 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (2 \, A a^{3} + 3 \, {\left (4 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{2} + {\left (4 \, B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt {b x + a}}{4 \, {\left (a^{4} b x^{3} + a^{5} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.31, size = 125, normalized size = 1.14 \begin {gather*} -\frac {3 \, {\left (4 \, B a b - 5 \, A b^{2}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a^{3}} - \frac {2 \, {\left (B a b - A b^{2}\right )}}{\sqrt {b x + a} a^{3}} - \frac {4 \, {\left (b x + a\right )}^{\frac {3}{2}} B a b - 4 \, \sqrt {b x + a} B a^{2} b - 7 \, {\left (b x + a\right )}^{\frac {3}{2}} A b^{2} + 9 \, \sqrt {b x + a} A a b^{2}}{4 \, a^{3} b^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 101, normalized size = 0.92 \begin {gather*} 2 \left (-\frac {-A b +B a}{\sqrt {b x +a}\, a^{3}}+\frac {-\frac {3 \left (5 A b -4 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 \sqrt {a}}+\frac {\left (\frac {7 A b}{8}-\frac {B a}{2}\right ) \left (b x +a \right )^{\frac {3}{2}}+\left (-\frac {9}{8} A a b +\frac {1}{2} B \,a^{2}\right ) \sqrt {b x +a}}{b^{2} x^{2}}}{a^{3}}\right ) b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.96, size = 144, normalized size = 1.31 \begin {gather*} -\frac {1}{8} \, b^{2} {\left (\frac {2 \, {\left (8 \, B a^{3} - 8 \, A a^{2} b + 3 \, {\left (4 \, B a - 5 \, A b\right )} {\left (b x + a\right )}^{2} - 5 \, {\left (4 \, B a^{2} - 5 \, A a b\right )} {\left (b x + a\right )}\right )}}{{\left (b x + a\right )}^{\frac {5}{2}} a^{3} b - 2 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} b + \sqrt {b x + a} a^{5} b} + \frac {3 \, {\left (4 \, B a - 5 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {7}{2}} b}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 123, normalized size = 1.12 \begin {gather*} \frac {\frac {2\,\left (A\,b^2-B\,a\,b\right )}{a}-\frac {5\,\left (5\,A\,b^2-4\,B\,a\,b\right )\,\left (a+b\,x\right )}{4\,a^2}+\frac {3\,\left (5\,A\,b^2-4\,B\,a\,b\right )\,{\left (a+b\,x\right )}^2}{4\,a^3}}{{\left (a+b\,x\right )}^{5/2}-2\,a\,{\left (a+b\,x\right )}^{3/2}+a^2\,\sqrt {a+b\,x}}-\frac {3\,b\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (5\,A\,b-4\,B\,a\right )}{4\,a^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 102.54, size = 185, normalized size = 1.68 \begin {gather*} A \left (- \frac {1}{2 a \sqrt {b} x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {5 \sqrt {b}}{4 a^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {15 b^{\frac {3}{2}}}{4 a^{3} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} - \frac {15 b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{4 a^{\frac {7}{2}}}\right ) + B \left (- \frac {1}{a \sqrt {b} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {3 \sqrt {b}}{a^{2} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {3 b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{a^{\frac {5}{2}}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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