Optimal. Leaf size=147 \[ -\frac {5 \sqrt {c} (3 b B-7 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 b^{9/2}}-\frac {5 (3 b B-7 A c)}{4 b^4 \sqrt {x}}+\frac {5 (3 b B-7 A c)}{12 b^3 c x^{3/2}}-\frac {3 b B-7 A c}{4 b^2 c x^{3/2} (b+c x)}-\frac {b B-A c}{2 b c x^{3/2} (b+c x)^2} \]
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Rubi [A] time = 0.07, antiderivative size = 147, 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 {3 b B-7 A c}{4 b^2 c x^{3/2} (b+c x)}+\frac {5 (3 b B-7 A c)}{12 b^3 c x^{3/2}}-\frac {5 (3 b B-7 A c)}{4 b^4 \sqrt {x}}-\frac {5 \sqrt {c} (3 b B-7 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 b^{9/2}}-\frac {b B-A c}{2 b c x^{3/2} (b+c x)^2} \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 {\sqrt {x} (A+B x)}{\left (b x+c x^2\right )^3} \, dx &=\int \frac {A+B x}{x^{5/2} (b+c x)^3} \, dx\\ &=-\frac {b B-A c}{2 b c x^{3/2} (b+c x)^2}-\frac {\left (\frac {3 b B}{2}-\frac {7 A c}{2}\right ) \int \frac {1}{x^{5/2} (b+c x)^2} \, dx}{2 b c}\\ &=-\frac {b B-A c}{2 b c x^{3/2} (b+c x)^2}-\frac {3 b B-7 A c}{4 b^2 c x^{3/2} (b+c x)}-\frac {(5 (3 b B-7 A c)) \int \frac {1}{x^{5/2} (b+c x)} \, dx}{8 b^2 c}\\ &=\frac {5 (3 b B-7 A c)}{12 b^3 c x^{3/2}}-\frac {b B-A c}{2 b c x^{3/2} (b+c x)^2}-\frac {3 b B-7 A c}{4 b^2 c x^{3/2} (b+c x)}+\frac {(5 (3 b B-7 A c)) \int \frac {1}{x^{3/2} (b+c x)} \, dx}{8 b^3}\\ &=\frac {5 (3 b B-7 A c)}{12 b^3 c x^{3/2}}-\frac {5 (3 b B-7 A c)}{4 b^4 \sqrt {x}}-\frac {b B-A c}{2 b c x^{3/2} (b+c x)^2}-\frac {3 b B-7 A c}{4 b^2 c x^{3/2} (b+c x)}-\frac {(5 c (3 b B-7 A c)) \int \frac {1}{\sqrt {x} (b+c x)} \, dx}{8 b^4}\\ &=\frac {5 (3 b B-7 A c)}{12 b^3 c x^{3/2}}-\frac {5 (3 b B-7 A c)}{4 b^4 \sqrt {x}}-\frac {b B-A c}{2 b c x^{3/2} (b+c x)^2}-\frac {3 b B-7 A c}{4 b^2 c x^{3/2} (b+c x)}-\frac {(5 c (3 b B-7 A c)) \operatorname {Subst}\left (\int \frac {1}{b+c x^2} \, dx,x,\sqrt {x}\right )}{4 b^4}\\ &=\frac {5 (3 b B-7 A c)}{12 b^3 c x^{3/2}}-\frac {5 (3 b B-7 A c)}{4 b^4 \sqrt {x}}-\frac {b B-A c}{2 b c x^{3/2} (b+c x)^2}-\frac {3 b B-7 A c}{4 b^2 c x^{3/2} (b+c x)}-\frac {5 \sqrt {c} (3 b B-7 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 b^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 61, normalized size = 0.41 \begin {gather*} \frac {\frac {3 b^2 (A c-b B)}{(b+c x)^2}+(3 b B-7 A c) \, _2F_1\left (-\frac {3}{2},2;-\frac {1}{2};-\frac {c x}{b}\right )}{6 b^3 c x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 125, normalized size = 0.85 \begin {gather*} \frac {-8 A b^3+56 A b^2 c x+175 A b c^2 x^2+105 A c^3 x^3-24 b^3 B x-75 b^2 B c x^2-45 b B c^2 x^3}{12 b^4 x^{3/2} (b+c x)^2}-\frac {5 \left (3 b B \sqrt {c}-7 A c^{3/2}\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 b^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 380, normalized size = 2.59 \begin {gather*} \left [-\frac {15 \, {\left ({\left (3 \, B b c^{2} - 7 \, A c^{3}\right )} x^{4} + 2 \, {\left (3 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{3} + {\left (3 \, B b^{3} - 7 \, A b^{2} 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 (8 \, A b^{3} + 15 \, {\left (3 \, B b c^{2} - 7 \, A c^{3}\right )} x^{3} + 25 \, {\left (3 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{2} + 8 \, {\left (3 \, B b^{3} - 7 \, A b^{2} c\right )} x\right )} \sqrt {x}}{24 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}}, \frac {15 \, {\left ({\left (3 \, B b c^{2} - 7 \, A c^{3}\right )} x^{4} + 2 \, {\left (3 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{3} + {\left (3 \, B b^{3} - 7 \, A b^{2} c\right )} x^{2}\right )} \sqrt {\frac {c}{b}} \arctan \left (\frac {b \sqrt {\frac {c}{b}}}{c \sqrt {x}}\right ) - {\left (8 \, A b^{3} + 15 \, {\left (3 \, B b c^{2} - 7 \, A c^{3}\right )} x^{3} + 25 \, {\left (3 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{2} + 8 \, {\left (3 \, B b^{3} - 7 \, A b^{2} c\right )} x\right )} \sqrt {x}}{12 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 108, normalized size = 0.73 \begin {gather*} -\frac {5 \, {\left (3 \, B b c - 7 \, A c^{2}\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \, \sqrt {b c} b^{4}} - \frac {2 \, {\left (3 \, B b x - 9 \, A c x + A b\right )}}{3 \, b^{4} x^{\frac {3}{2}}} - \frac {7 \, B b c^{2} x^{\frac {3}{2}} - 11 \, A c^{3} x^{\frac {3}{2}} + 9 \, B b^{2} c \sqrt {x} - 13 \, A b c^{2} \sqrt {x}}{4 \, {\left (c x + b\right )}^{2} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 152, normalized size = 1.03 \begin {gather*} \frac {11 A \,c^{3} x^{\frac {3}{2}}}{4 \left (c x +b \right )^{2} b^{4}}-\frac {7 B \,c^{2} x^{\frac {3}{2}}}{4 \left (c x +b \right )^{2} b^{3}}+\frac {13 A \,c^{2} \sqrt {x}}{4 \left (c x +b \right )^{2} b^{3}}-\frac {9 B c \sqrt {x}}{4 \left (c x +b \right )^{2} b^{2}}+\frac {35 A \,c^{2} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \sqrt {b c}\, b^{4}}-\frac {15 B c \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \sqrt {b c}\, b^{3}}+\frac {6 A c}{b^{4} \sqrt {x}}-\frac {2 B}{b^{3} \sqrt {x}}-\frac {2 A}{3 b^{3} x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 128, normalized size = 0.87 \begin {gather*} -\frac {8 \, A b^{3} + 15 \, {\left (3 \, B b c^{2} - 7 \, A c^{3}\right )} x^{3} + 25 \, {\left (3 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{2} + 8 \, {\left (3 \, B b^{3} - 7 \, A b^{2} c\right )} x}{12 \, {\left (b^{4} c^{2} x^{\frac {7}{2}} + 2 \, b^{5} c x^{\frac {5}{2}} + b^{6} x^{\frac {3}{2}}\right )}} - \frac {5 \, {\left (3 \, B b c - 7 \, A c^{2}\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \, \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 = 114, normalized size = 0.78 \begin {gather*} \frac {\frac {2\,x\,\left (7\,A\,c-3\,B\,b\right )}{3\,b^2}-\frac {2\,A}{3\,b}+\frac {5\,c^2\,x^3\,\left (7\,A\,c-3\,B\,b\right )}{4\,b^4}+\frac {25\,c\,x^2\,\left (7\,A\,c-3\,B\,b\right )}{12\,b^3}}{b^2\,x^{3/2}+c^2\,x^{7/2}+2\,b\,c\,x^{5/2}}+\frac {5\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {x}}{\sqrt {b}}\right )\,\left (7\,A\,c-3\,B\,b\right )}{4\,b^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 165.44, size = 1880, normalized size = 12.79
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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