Optimal. Leaf size=131 \[ -\frac {b^{3/2} (7 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{9/2}}-\frac {b (7 A b-5 a B)}{a^4 \sqrt {x}}+\frac {7 A b-5 a B}{3 a^3 x^{3/2}}-\frac {7 A b-5 a B}{5 a^2 b x^{5/2}}+\frac {A b-a B}{a b x^{5/2} (a+b x)} \]
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Rubi [A] time = 0.06, antiderivative size = 131, normalized size of antiderivative = 1.00, 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, 205} \begin {gather*} -\frac {b^{3/2} (7 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{9/2}}+\frac {7 A b-5 a B}{3 a^3 x^{3/2}}-\frac {7 A b-5 a B}{5 a^2 b x^{5/2}}-\frac {b (7 A b-5 a B)}{a^4 \sqrt {x}}+\frac {A b-a B}{a b x^{5/2} (a+b x)} \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^{7/2} (a+b x)^2} \, dx &=\frac {A b-a B}{a b x^{5/2} (a+b x)}-\frac {\left (-\frac {7 A b}{2}+\frac {5 a B}{2}\right ) \int \frac {1}{x^{7/2} (a+b x)} \, dx}{a b}\\ &=-\frac {7 A b-5 a B}{5 a^2 b x^{5/2}}+\frac {A b-a B}{a b x^{5/2} (a+b x)}-\frac {(7 A b-5 a B) \int \frac {1}{x^{5/2} (a+b x)} \, dx}{2 a^2}\\ &=-\frac {7 A b-5 a B}{5 a^2 b x^{5/2}}+\frac {7 A b-5 a B}{3 a^3 x^{3/2}}+\frac {A b-a B}{a b x^{5/2} (a+b x)}+\frac {(b (7 A b-5 a B)) \int \frac {1}{x^{3/2} (a+b x)} \, dx}{2 a^3}\\ &=-\frac {7 A b-5 a B}{5 a^2 b x^{5/2}}+\frac {7 A b-5 a B}{3 a^3 x^{3/2}}-\frac {b (7 A b-5 a B)}{a^4 \sqrt {x}}+\frac {A b-a B}{a b x^{5/2} (a+b x)}-\frac {\left (b^2 (7 A b-5 a B)\right ) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{2 a^4}\\ &=-\frac {7 A b-5 a B}{5 a^2 b x^{5/2}}+\frac {7 A b-5 a B}{3 a^3 x^{3/2}}-\frac {b (7 A b-5 a B)}{a^4 \sqrt {x}}+\frac {A b-a B}{a b x^{5/2} (a+b x)}-\frac {\left (b^2 (7 A b-5 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{a^4}\\ &=-\frac {7 A b-5 a B}{5 a^2 b x^{5/2}}+\frac {7 A b-5 a B}{3 a^3 x^{3/2}}-\frac {b (7 A b-5 a B)}{a^4 \sqrt {x}}+\frac {A b-a B}{a b x^{5/2} (a+b x)}-\frac {b^{3/2} (7 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 64, normalized size = 0.49 \begin {gather*} \frac {(a+b x) (5 a B-7 A b) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};-\frac {b x}{a}\right )+5 a (A b-a B)}{5 a^2 b x^{5/2} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 122, normalized size = 0.93 \begin {gather*} \frac {\left (5 a b^{3/2} B-7 A b^{5/2}\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{9/2}}+\frac {-6 a^3 A-10 a^3 B x+14 a^2 A b x+50 a^2 b B x^2-70 a A b^2 x^2+75 a b^2 B x^3-105 A b^3 x^3}{15 a^4 x^{5/2} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.14, size = 319, normalized size = 2.44 \begin {gather*} \left [-\frac {15 \, {\left ({\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{4} + {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x - 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (6 \, A a^{3} - 15 \, {\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} - 10 \, {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 2 \, {\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt {x}}{30 \, {\left (a^{4} b x^{4} + a^{5} x^{3}\right )}}, -\frac {15 \, {\left ({\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{4} + {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) + {\left (6 \, A a^{3} - 15 \, {\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} - 10 \, {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 2 \, {\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt {x}}{15 \, {\left (a^{4} b x^{4} + a^{5} x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.23, size = 110, normalized size = 0.84 \begin {gather*} \frac {{\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} + \frac {B a b^{2} \sqrt {x} - A b^{3} \sqrt {x}}{{\left (b x + a\right )} a^{4}} + \frac {2 \, {\left (30 \, B a b x^{2} - 45 \, A b^{2} x^{2} - 5 \, B a^{2} x + 10 \, A a b x - 3 \, A a^{2}\right )}}{15 \, a^{4} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 139, normalized size = 1.06 \begin {gather*} -\frac {7 A \,b^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{4}}+\frac {5 B \,b^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{3}}-\frac {A \,b^{3} \sqrt {x}}{\left (b x +a \right ) a^{4}}+\frac {B \,b^{2} \sqrt {x}}{\left (b x +a \right ) a^{3}}-\frac {6 A \,b^{2}}{a^{4} \sqrt {x}}+\frac {4 B b}{a^{3} \sqrt {x}}+\frac {4 A b}{3 a^{3} x^{\frac {3}{2}}}-\frac {2 B}{3 a^{2} x^{\frac {3}{2}}}-\frac {2 A}{5 a^{2} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.99, size = 118, normalized size = 0.90 \begin {gather*} -\frac {6 \, A a^{3} - 15 \, {\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} - 10 \, {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 2 \, {\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x}{15 \, {\left (a^{4} b x^{\frac {7}{2}} + a^{5} x^{\frac {5}{2}}\right )}} + \frac {{\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 103, normalized size = 0.79 \begin {gather*} -\frac {\frac {2\,A}{5\,a}-\frac {2\,x\,\left (7\,A\,b-5\,B\,a\right )}{15\,a^2}+\frac {b^2\,x^3\,\left (7\,A\,b-5\,B\,a\right )}{a^4}+\frac {2\,b\,x^2\,\left (7\,A\,b-5\,B\,a\right )}{3\,a^3}}{a\,x^{5/2}+b\,x^{7/2}}-\frac {b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (7\,A\,b-5\,B\,a\right )}{a^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 161.12, size = 1127, normalized size = 8.60
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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