Optimal. Leaf size=136 \[ -\frac {5 b (7 A b-4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 a^{9/2}}+\frac {5 b (7 A b-4 a B)}{4 a^4 \sqrt {a+b x}}+\frac {5 b (7 A b-4 a B)}{12 a^3 (a+b x)^{3/2}}+\frac {7 A b-4 a B}{4 a^2 x (a+b x)^{3/2}}-\frac {A}{2 a x^2 (a+b x)^{3/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 140, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {78, 51, 63, 208} \[ \frac {5 \sqrt {a+b x} (7 A b-4 a B)}{4 a^4 x}-\frac {5 (7 A b-4 a B)}{6 a^3 x \sqrt {a+b x}}-\frac {7 A b-4 a B}{6 a^2 x (a+b x)^{3/2}}-\frac {5 b (7 A b-4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 a^{9/2}}-\frac {A}{2 a x^2 (a+b x)^{3/2}} \]
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)^{5/2}} \, dx &=-\frac {A}{2 a x^2 (a+b x)^{3/2}}+\frac {\left (-\frac {7 A b}{2}+2 a B\right ) \int \frac {1}{x^2 (a+b x)^{5/2}} \, dx}{2 a}\\ &=-\frac {A}{2 a x^2 (a+b x)^{3/2}}-\frac {7 A b-4 a B}{6 a^2 x (a+b x)^{3/2}}-\frac {(5 (7 A b-4 a B)) \int \frac {1}{x^2 (a+b x)^{3/2}} \, dx}{12 a^2}\\ &=-\frac {A}{2 a x^2 (a+b x)^{3/2}}-\frac {7 A b-4 a B}{6 a^2 x (a+b x)^{3/2}}-\frac {5 (7 A b-4 a B)}{6 a^3 x \sqrt {a+b x}}-\frac {(5 (7 A b-4 a B)) \int \frac {1}{x^2 \sqrt {a+b x}} \, dx}{4 a^3}\\ &=-\frac {A}{2 a x^2 (a+b x)^{3/2}}-\frac {7 A b-4 a B}{6 a^2 x (a+b x)^{3/2}}-\frac {5 (7 A b-4 a B)}{6 a^3 x \sqrt {a+b x}}+\frac {5 (7 A b-4 a B) \sqrt {a+b x}}{4 a^4 x}+\frac {(5 b (7 A b-4 a B)) \int \frac {1}{x \sqrt {a+b x}} \, dx}{8 a^4}\\ &=-\frac {A}{2 a x^2 (a+b x)^{3/2}}-\frac {7 A b-4 a B}{6 a^2 x (a+b x)^{3/2}}-\frac {5 (7 A b-4 a B)}{6 a^3 x \sqrt {a+b x}}+\frac {5 (7 A b-4 a B) \sqrt {a+b x}}{4 a^4 x}+\frac {(5 (7 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^4}\\ &=-\frac {A}{2 a x^2 (a+b x)^{3/2}}-\frac {7 A b-4 a B}{6 a^2 x (a+b x)^{3/2}}-\frac {5 (7 A b-4 a B)}{6 a^3 x \sqrt {a+b x}}+\frac {5 (7 A b-4 a B) \sqrt {a+b x}}{4 a^4 x}-\frac {5 b (7 A b-4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 a^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 56, normalized size = 0.41 \[ \frac {b x^2 (7 A b-4 a B) \, _2F_1\left (-\frac {3}{2},2;-\frac {1}{2};\frac {b x}{a}+1\right )-3 a^2 A}{6 a^3 x^2 (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 395, normalized size = 2.90 \[ \left [-\frac {15 \, {\left ({\left (4 \, B a b^{3} - 7 \, A b^{4}\right )} x^{4} + 2 \, {\left (4 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{3} + {\left (4 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {a} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (6 \, A a^{4} + 15 \, {\left (4 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{3} + 20 \, {\left (4 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{2} + 3 \, {\left (4 \, B a^{4} - 7 \, A a^{3} b\right )} x\right )} \sqrt {b x + a}}{24 \, {\left (a^{5} b^{2} x^{4} + 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}}, -\frac {15 \, {\left ({\left (4 \, B a b^{3} - 7 \, A b^{4}\right )} x^{4} + 2 \, {\left (4 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{3} + {\left (4 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (6 \, A a^{4} + 15 \, {\left (4 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{3} + 20 \, {\left (4 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{2} + 3 \, {\left (4 \, B a^{4} - 7 \, A a^{3} b\right )} x\right )} \sqrt {b x + a}}{12 \, {\left (a^{5} b^{2} x^{4} + 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.33, size = 149, normalized size = 1.10 \[ -\frac {5 \, {\left (4 \, B a b - 7 \, A b^{2}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a^{4}} - \frac {2 \, {\left (6 \, {\left (b x + a\right )} B a b + B a^{2} b - 9 \, {\left (b x + a\right )} A b^{2} - A a b^{2}\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4}} - \frac {4 \, {\left (b x + a\right )}^{\frac {3}{2}} B a b - 4 \, \sqrt {b x + a} B a^{2} b - 11 \, {\left (b x + a\right )}^{\frac {3}{2}} A b^{2} + 13 \, \sqrt {b x + a} A a b^{2}}{4 \, a^{4} b^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 122, normalized size = 0.90 \[ 2 \left (-\frac {-A b +B a}{3 \left (b x +a \right )^{\frac {3}{2}} a^{3}}-\frac {-3 A b +2 B a}{\sqrt {b x +a}\, a^{4}}+\frac {-\frac {5 \left (7 A b -4 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 \sqrt {a}}+\frac {\left (\frac {11 A b}{8}-\frac {B a}{2}\right ) \left (b x +a \right )^{\frac {3}{2}}+\left (-\frac {13}{8} A a b +\frac {1}{2} B \,a^{2}\right ) \sqrt {b x +a}}{b^{2} x^{2}}}{a^{4}}\right ) b \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.97, size = 167, normalized size = 1.23 \[ -\frac {1}{24} \, b^{2} {\left (\frac {2 \, {\left (8 \, B a^{4} - 8 \, A a^{3} b + 15 \, {\left (4 \, B a - 7 \, A b\right )} {\left (b x + a\right )}^{3} - 25 \, {\left (4 \, B a^{2} - 7 \, A a b\right )} {\left (b x + a\right )}^{2} + 8 \, {\left (4 \, B a^{3} - 7 \, A a^{2} b\right )} {\left (b x + a\right )}\right )}}{{\left (b x + a\right )}^{\frac {7}{2}} a^{4} b - 2 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{5} b + {\left (b x + a\right )}^{\frac {3}{2}} a^{6} b} + \frac {15 \, {\left (4 \, B a - 7 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {9}{2}} b}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 147, normalized size = 1.08 \[ \frac {\frac {2\,\left (A\,b^2-B\,a\,b\right )}{3\,a}+\frac {2\,\left (7\,A\,b^2-4\,B\,a\,b\right )\,\left (a+b\,x\right )}{3\,a^2}-\frac {25\,\left (7\,A\,b^2-4\,B\,a\,b\right )\,{\left (a+b\,x\right )}^2}{12\,a^3}+\frac {5\,\left (7\,A\,b^2-4\,B\,a\,b\right )\,{\left (a+b\,x\right )}^3}{4\,a^4}}{{\left (a+b\,x\right )}^{7/2}-2\,a\,{\left (a+b\,x\right )}^{5/2}+a^2\,{\left (a+b\,x\right )}^{3/2}}-\frac {5\,b\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (7\,A\,b-4\,B\,a\right )}{4\,a^{9/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 122.38, size = 1287, normalized size = 9.46 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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