Optimal. Leaf size=130 \[ \frac {c^{3/2} (5 b B-7 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{9/2}}+\frac {c (5 b B-7 A c)}{b^4 \sqrt {x}}-\frac {5 b B-7 A c}{3 b^3 x^{3/2}}+\frac {5 b B-7 A c}{5 b^2 c x^{5/2}}-\frac {b B-A c}{b c x^{5/2} (b+c x)} \]
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Rubi [A] time = 0.07, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {781, 78, 51, 63, 205} \begin {gather*} \frac {c^{3/2} (5 b B-7 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{9/2}}-\frac {5 b B-7 A c}{3 b^3 x^{3/2}}+\frac {5 b B-7 A c}{5 b^2 c x^{5/2}}+\frac {c (5 b B-7 A c)}{b^4 \sqrt {x}}-\frac {b B-A c}{b c x^{5/2} (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 205
Rule 781
Rubi steps
\begin {align*} \int \frac {A+B x}{x^{3/2} \left (b x+c x^2\right )^2} \, dx &=\int \frac {A+B x}{x^{7/2} (b+c x)^2} \, dx\\ &=-\frac {b B-A c}{b c x^{5/2} (b+c x)}-\frac {\left (\frac {5 b B}{2}-\frac {7 A c}{2}\right ) \int \frac {1}{x^{7/2} (b+c x)} \, dx}{b c}\\ &=\frac {5 b B-7 A c}{5 b^2 c x^{5/2}}-\frac {b B-A c}{b c x^{5/2} (b+c x)}+\frac {(5 b B-7 A c) \int \frac {1}{x^{5/2} (b+c x)} \, dx}{2 b^2}\\ &=\frac {5 b B-7 A c}{5 b^2 c x^{5/2}}-\frac {5 b B-7 A c}{3 b^3 x^{3/2}}-\frac {b B-A c}{b c x^{5/2} (b+c x)}-\frac {(c (5 b B-7 A c)) \int \frac {1}{x^{3/2} (b+c x)} \, dx}{2 b^3}\\ &=\frac {5 b B-7 A c}{5 b^2 c x^{5/2}}-\frac {5 b B-7 A c}{3 b^3 x^{3/2}}+\frac {c (5 b B-7 A c)}{b^4 \sqrt {x}}-\frac {b B-A c}{b c x^{5/2} (b+c x)}+\frac {\left (c^2 (5 b B-7 A c)\right ) \int \frac {1}{\sqrt {x} (b+c x)} \, dx}{2 b^4}\\ &=\frac {5 b B-7 A c}{5 b^2 c x^{5/2}}-\frac {5 b B-7 A c}{3 b^3 x^{3/2}}+\frac {c (5 b B-7 A c)}{b^4 \sqrt {x}}-\frac {b B-A c}{b c x^{5/2} (b+c x)}+\frac {\left (c^2 (5 b B-7 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{b+c x^2} \, dx,x,\sqrt {x}\right )}{b^4}\\ &=\frac {5 b B-7 A c}{5 b^2 c x^{5/2}}-\frac {5 b B-7 A c}{3 b^3 x^{3/2}}+\frac {c (5 b B-7 A c)}{b^4 \sqrt {x}}-\frac {b B-A c}{b c x^{5/2} (b+c x)}+\frac {c^{3/2} (5 b B-7 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 64, normalized size = 0.49 \begin {gather*} \frac {(b+c x) (5 b B-7 A c) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};-\frac {c x}{b}\right )+5 b (A c-b B)}{5 b^2 c x^{5/2} (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 122, normalized size = 0.94 \begin {gather*} \frac {\left (5 b B c^{3/2}-7 A c^{5/2}\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{9/2}}+\frac {-6 A b^3+14 A b^2 c x-70 A b c^2 x^2-105 A c^3 x^3-10 b^3 B x+50 b^2 B c x^2+75 b B c^2 x^3}{15 b^4 x^{5/2} (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 319, normalized size = 2.45 \begin {gather*} \left [-\frac {15 \, {\left ({\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} x^{4} + {\left (5 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{3}\right )} \sqrt {-\frac {c}{b}} \log \left (\frac {c x - 2 \, b \sqrt {x} \sqrt {-\frac {c}{b}} - b}{c x + b}\right ) + 2 \, {\left (6 \, A b^{3} - 15 \, {\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} x^{3} - 10 \, {\left (5 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{2} + 2 \, {\left (5 \, B b^{3} - 7 \, A b^{2} c\right )} x\right )} \sqrt {x}}{30 \, {\left (b^{4} c x^{4} + b^{5} x^{3}\right )}}, -\frac {15 \, {\left ({\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} x^{4} + {\left (5 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{3}\right )} \sqrt {\frac {c}{b}} \arctan \left (\frac {b \sqrt {\frac {c}{b}}}{c \sqrt {x}}\right ) + {\left (6 \, A b^{3} - 15 \, {\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} x^{3} - 10 \, {\left (5 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{2} + 2 \, {\left (5 \, B b^{3} - 7 \, A b^{2} c\right )} x\right )} \sqrt {x}}{15 \, {\left (b^{4} c x^{4} + b^{5} x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 110, normalized size = 0.85 \begin {gather*} \frac {{\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} b^{4}} + \frac {B b c^{2} \sqrt {x} - A c^{3} \sqrt {x}}{{\left (c x + b\right )} b^{4}} + \frac {2 \, {\left (30 \, B b c x^{2} - 45 \, A c^{2} x^{2} - 5 \, B b^{2} x + 10 \, A b c x - 3 \, A b^{2}\right )}}{15 \, b^{4} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 139, normalized size = 1.07 \begin {gather*} -\frac {7 A \,c^{3} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, b^{4}}+\frac {5 B \,c^{2} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, b^{3}}-\frac {A \,c^{3} \sqrt {x}}{\left (c x +b \right ) b^{4}}+\frac {B \,c^{2} \sqrt {x}}{\left (c x +b \right ) b^{3}}-\frac {6 A \,c^{2}}{b^{4} \sqrt {x}}+\frac {4 B c}{b^{3} \sqrt {x}}+\frac {4 A c}{3 b^{3} x^{\frac {3}{2}}}-\frac {2 B}{3 b^{2} x^{\frac {3}{2}}}-\frac {2 A}{5 b^{2} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.17, size = 118, normalized size = 0.91 \begin {gather*} -\frac {6 \, A b^{3} - 15 \, {\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} x^{3} - 10 \, {\left (5 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{2} + 2 \, {\left (5 \, B b^{3} - 7 \, A b^{2} c\right )} x}{15 \, {\left (b^{4} c x^{\frac {7}{2}} + b^{5} x^{\frac {5}{2}}\right )}} + \frac {{\left (5 \, B b c^{2} - 7 \, A c^{3}\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.11, size = 103, normalized size = 0.79 \begin {gather*} -\frac {\frac {2\,A}{5\,b}-\frac {2\,x\,\left (7\,A\,c-5\,B\,b\right )}{15\,b^2}+\frac {c^2\,x^3\,\left (7\,A\,c-5\,B\,b\right )}{b^4}+\frac {2\,c\,x^2\,\left (7\,A\,c-5\,B\,b\right )}{3\,b^3}}{b\,x^{5/2}+c\,x^{7/2}}-\frac {c^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {x}}{\sqrt {b}}\right )\,\left (7\,A\,c-5\,B\,b\right )}{b^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 81.93, size = 1127, normalized size = 8.67
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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