Optimal. Leaf size=110 \[ -\frac {\sqrt {c} (3 b B-5 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{7/2}}-\frac {3 b B-5 A c}{b^3 \sqrt {x}}+\frac {3 b B-5 A c}{3 b^2 c x^{3/2}}-\frac {b B-A c}{b c x^{3/2} (b+c x)} \]
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Rubi [A] time = 0.06, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {3 b B-5 A c}{3 b^2 c x^{3/2}}-\frac {3 b B-5 A c}{b^3 \sqrt {x}}-\frac {\sqrt {c} (3 b B-5 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{7/2}}-\frac {b B-A c}{b c x^{3/2} (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}{\sqrt {x} \left (b x+c x^2\right )^2} \, dx &=\int \frac {A+B x}{x^{5/2} (b+c x)^2} \, dx\\ &=-\frac {b B-A c}{b c x^{3/2} (b+c x)}-\frac {\left (\frac {3 b B}{2}-\frac {5 A c}{2}\right ) \int \frac {1}{x^{5/2} (b+c x)} \, dx}{b c}\\ &=\frac {3 b B-5 A c}{3 b^2 c x^{3/2}}-\frac {b B-A c}{b c x^{3/2} (b+c x)}+\frac {(3 b B-5 A c) \int \frac {1}{x^{3/2} (b+c x)} \, dx}{2 b^2}\\ &=\frac {3 b B-5 A c}{3 b^2 c x^{3/2}}-\frac {3 b B-5 A c}{b^3 \sqrt {x}}-\frac {b B-A c}{b c x^{3/2} (b+c x)}-\frac {(c (3 b B-5 A c)) \int \frac {1}{\sqrt {x} (b+c x)} \, dx}{2 b^3}\\ &=\frac {3 b B-5 A c}{3 b^2 c x^{3/2}}-\frac {3 b B-5 A c}{b^3 \sqrt {x}}-\frac {b B-A c}{b c x^{3/2} (b+c x)}-\frac {(c (3 b B-5 A c)) \operatorname {Subst}\left (\int \frac {1}{b+c x^2} \, dx,x,\sqrt {x}\right )}{b^3}\\ &=\frac {3 b B-5 A c}{3 b^2 c x^{3/2}}-\frac {3 b B-5 A c}{b^3 \sqrt {x}}-\frac {b B-A c}{b c x^{3/2} (b+c x)}-\frac {\sqrt {c} (3 b B-5 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 64, normalized size = 0.58 \begin {gather*} \frac {(b+c x) (3 b B-5 A c) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\frac {c x}{b}\right )+3 b (A c-b B)}{3 b^2 c x^{3/2} (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 98, normalized size = 0.89 \begin {gather*} \frac {\left (5 A c^{3/2}-3 b B \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{7/2}}+\frac {-2 A b^2+10 A b c x+15 A c^2 x^2-6 b^2 B x-9 b B c x^2}{3 b^3 x^{3/2} (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 262, normalized size = 2.38 \begin {gather*} \left [-\frac {3 \, {\left ({\left (3 \, B b c - 5 \, A c^{2}\right )} x^{3} + {\left (3 \, B b^{2} - 5 \, A b c\right )} x^{2}\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 (2 \, A b^{2} + 3 \, {\left (3 \, B b c - 5 \, A c^{2}\right )} x^{2} + 2 \, {\left (3 \, B b^{2} - 5 \, A b c\right )} x\right )} \sqrt {x}}{6 \, {\left (b^{3} c x^{3} + b^{4} x^{2}\right )}}, \frac {3 \, {\left ({\left (3 \, B b c - 5 \, A c^{2}\right )} x^{3} + {\left (3 \, B b^{2} - 5 \, A b c\right )} x^{2}\right )} \sqrt {\frac {c}{b}} \arctan \left (\frac {b \sqrt {\frac {c}{b}}}{c \sqrt {x}}\right ) - {\left (2 \, A b^{2} + 3 \, {\left (3 \, B b c - 5 \, A c^{2}\right )} x^{2} + 2 \, {\left (3 \, B b^{2} - 5 \, A b c\right )} x\right )} \sqrt {x}}{3 \, {\left (b^{3} c x^{3} + b^{4} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 85, normalized size = 0.77 \begin {gather*} -\frac {{\left (3 \, B b c - 5 \, A c^{2}\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} b^{3}} - \frac {B b c \sqrt {x} - A c^{2} \sqrt {x}}{{\left (c x + b\right )} b^{3}} - \frac {2 \, {\left (3 \, B b x - 6 \, A c x + A b\right )}}{3 \, b^{3} x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 113, normalized size = 1.03 \begin {gather*} \frac {5 A \,c^{2} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, b^{3}}-\frac {3 B c \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, b^{2}}+\frac {A \,c^{2} \sqrt {x}}{\left (c x +b \right ) b^{3}}-\frac {B c \sqrt {x}}{\left (c x +b \right ) b^{2}}+\frac {4 A c}{b^{3} \sqrt {x}}-\frac {2 B}{b^{2} \sqrt {x}}-\frac {2 A}{3 b^{2} x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 93, normalized size = 0.85 \begin {gather*} -\frac {2 \, A b^{2} + 3 \, {\left (3 \, B b c - 5 \, A c^{2}\right )} x^{2} + 2 \, {\left (3 \, B b^{2} - 5 \, A b c\right )} x}{3 \, {\left (b^{3} c x^{\frac {5}{2}} + b^{4} x^{\frac {3}{2}}\right )}} - \frac {{\left (3 \, B b c - 5 \, A c^{2}\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 81, normalized size = 0.74 \begin {gather*} \frac {\frac {2\,x\,\left (5\,A\,c-3\,B\,b\right )}{3\,b^2}-\frac {2\,A}{3\,b}+\frac {c\,x^2\,\left (5\,A\,c-3\,B\,b\right )}{b^3}}{b\,x^{3/2}+c\,x^{5/2}}+\frac {\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {x}}{\sqrt {b}}\right )\,\left (5\,A\,c-3\,B\,b\right )}{b^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 42.92, size = 983, normalized size = 8.94 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{5 x^{\frac {5}{2}}}\right ) & \text {for}\: b = 0 \wedge c = 0 \\\frac {- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{5 x^{\frac {5}{2}}}}{c^{2}} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{3 x^{\frac {3}{2}}} - \frac {2 B}{\sqrt {x}}}{b^{2}} & \text {for}\: c = 0 \\- \frac {4 i A b^{\frac {5}{2}} \sqrt {\frac {1}{c}}}{6 i b^{\frac {9}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{c}} + 6 i b^{\frac {7}{2}} c x^{\frac {5}{2}} \sqrt {\frac {1}{c}}} + \frac {20 i A b^{\frac {3}{2}} c x \sqrt {\frac {1}{c}}}{6 i b^{\frac {9}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{c}} + 6 i b^{\frac {7}{2}} c x^{\frac {5}{2}} \sqrt {\frac {1}{c}}} + \frac {30 i A \sqrt {b} c^{2} x^{2} \sqrt {\frac {1}{c}}}{6 i b^{\frac {9}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{c}} + 6 i b^{\frac {7}{2}} c x^{\frac {5}{2}} \sqrt {\frac {1}{c}}} + \frac {15 A b c x^{\frac {3}{2}} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{6 i b^{\frac {9}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{c}} + 6 i b^{\frac {7}{2}} c x^{\frac {5}{2}} \sqrt {\frac {1}{c}}} - \frac {15 A b c x^{\frac {3}{2}} \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{6 i b^{\frac {9}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{c}} + 6 i b^{\frac {7}{2}} c x^{\frac {5}{2}} \sqrt {\frac {1}{c}}} + \frac {15 A c^{2} x^{\frac {5}{2}} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{6 i b^{\frac {9}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{c}} + 6 i b^{\frac {7}{2}} c x^{\frac {5}{2}} \sqrt {\frac {1}{c}}} - \frac {15 A c^{2} x^{\frac {5}{2}} \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{6 i b^{\frac {9}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{c}} + 6 i b^{\frac {7}{2}} c x^{\frac {5}{2}} \sqrt {\frac {1}{c}}} - \frac {12 i B b^{\frac {5}{2}} x \sqrt {\frac {1}{c}}}{6 i b^{\frac {9}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{c}} + 6 i b^{\frac {7}{2}} c x^{\frac {5}{2}} \sqrt {\frac {1}{c}}} - \frac {18 i B b^{\frac {3}{2}} c x^{2} \sqrt {\frac {1}{c}}}{6 i b^{\frac {9}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{c}} + 6 i b^{\frac {7}{2}} c x^{\frac {5}{2}} \sqrt {\frac {1}{c}}} - \frac {9 B b^{2} x^{\frac {3}{2}} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{6 i b^{\frac {9}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{c}} + 6 i b^{\frac {7}{2}} c x^{\frac {5}{2}} \sqrt {\frac {1}{c}}} + \frac {9 B b^{2} x^{\frac {3}{2}} \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{6 i b^{\frac {9}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{c}} + 6 i b^{\frac {7}{2}} c x^{\frac {5}{2}} \sqrt {\frac {1}{c}}} - \frac {9 B b c x^{\frac {5}{2}} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{6 i b^{\frac {9}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{c}} + 6 i b^{\frac {7}{2}} c x^{\frac {5}{2}} \sqrt {\frac {1}{c}}} + \frac {9 B b c x^{\frac {5}{2}} \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{6 i b^{\frac {9}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{c}} + 6 i b^{\frac {7}{2}} c x^{\frac {5}{2}} \sqrt {\frac {1}{c}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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