Optimal. Leaf size=214 \[ -\frac {35 c^2 (2 b B-3 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{8 b^{11/2}}+\frac {35 c^2 \sqrt {x} (2 b B-3 A c)}{8 b^5 \sqrt {b x+c x^2}}+\frac {35 c (2 b B-3 A c)}{24 b^4 \sqrt {x} \sqrt {b x+c x^2}}-\frac {7 c \sqrt {x} (2 b B-3 A c)}{12 b^3 \left (b x+c x^2\right )^{3/2}}-\frac {2 b B-3 A c}{4 b^2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}-\frac {A}{3 b x^{3/2} \left (b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.18, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {792, 672, 666, 660, 207} \begin {gather*} \frac {35 c^2 \sqrt {x} (2 b B-3 A c)}{8 b^5 \sqrt {b x+c x^2}}-\frac {35 c^2 (2 b B-3 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{8 b^{11/2}}+\frac {35 c (2 b B-3 A c)}{24 b^4 \sqrt {x} \sqrt {b x+c x^2}}-\frac {7 c \sqrt {x} (2 b B-3 A c)}{12 b^3 \left (b x+c x^2\right )^{3/2}}-\frac {2 b B-3 A c}{4 b^2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}-\frac {A}{3 b x^{3/2} \left (b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 207
Rule 660
Rule 666
Rule 672
Rule 792
Rubi steps
\begin {align*} \int \frac {A+B x}{x^{3/2} \left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {A}{3 b x^{3/2} \left (b x+c x^2\right )^{3/2}}--\frac {\left (-\frac {3}{2} (-b B+A c)-\frac {3}{2} (-b B+2 A c)\right ) \int \frac {1}{\sqrt {x} \left (b x+c x^2\right )^{5/2}} \, dx}{3 b}\\ &=-\frac {A}{3 b x^{3/2} \left (b x+c x^2\right )^{3/2}}-\frac {2 b B-3 A c}{4 b^2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}-\frac {(7 c (2 b B-3 A c)) \int \frac {\sqrt {x}}{\left (b x+c x^2\right )^{5/2}} \, dx}{8 b^2}\\ &=-\frac {A}{3 b x^{3/2} \left (b x+c x^2\right )^{3/2}}-\frac {2 b B-3 A c}{4 b^2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}-\frac {7 c (2 b B-3 A c) \sqrt {x}}{12 b^3 \left (b x+c x^2\right )^{3/2}}-\frac {(35 c (2 b B-3 A c)) \int \frac {1}{\sqrt {x} \left (b x+c x^2\right )^{3/2}} \, dx}{24 b^3}\\ &=-\frac {A}{3 b x^{3/2} \left (b x+c x^2\right )^{3/2}}-\frac {2 b B-3 A c}{4 b^2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}-\frac {7 c (2 b B-3 A c) \sqrt {x}}{12 b^3 \left (b x+c x^2\right )^{3/2}}+\frac {35 c (2 b B-3 A c)}{24 b^4 \sqrt {x} \sqrt {b x+c x^2}}+\frac {\left (35 c^2 (2 b B-3 A c)\right ) \int \frac {\sqrt {x}}{\left (b x+c x^2\right )^{3/2}} \, dx}{16 b^4}\\ &=-\frac {A}{3 b x^{3/2} \left (b x+c x^2\right )^{3/2}}-\frac {2 b B-3 A c}{4 b^2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}-\frac {7 c (2 b B-3 A c) \sqrt {x}}{12 b^3 \left (b x+c x^2\right )^{3/2}}+\frac {35 c (2 b B-3 A c)}{24 b^4 \sqrt {x} \sqrt {b x+c x^2}}+\frac {35 c^2 (2 b B-3 A c) \sqrt {x}}{8 b^5 \sqrt {b x+c x^2}}+\frac {\left (35 c^2 (2 b B-3 A c)\right ) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{16 b^5}\\ &=-\frac {A}{3 b x^{3/2} \left (b x+c x^2\right )^{3/2}}-\frac {2 b B-3 A c}{4 b^2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}-\frac {7 c (2 b B-3 A c) \sqrt {x}}{12 b^3 \left (b x+c x^2\right )^{3/2}}+\frac {35 c (2 b B-3 A c)}{24 b^4 \sqrt {x} \sqrt {b x+c x^2}}+\frac {35 c^2 (2 b B-3 A c) \sqrt {x}}{8 b^5 \sqrt {b x+c x^2}}+\frac {\left (35 c^2 (2 b B-3 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )}{8 b^5}\\ &=-\frac {A}{3 b x^{3/2} \left (b x+c x^2\right )^{3/2}}-\frac {2 b B-3 A c}{4 b^2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}-\frac {7 c (2 b B-3 A c) \sqrt {x}}{12 b^3 \left (b x+c x^2\right )^{3/2}}+\frac {35 c (2 b B-3 A c)}{24 b^4 \sqrt {x} \sqrt {b x+c x^2}}+\frac {35 c^2 (2 b B-3 A c) \sqrt {x}}{8 b^5 \sqrt {b x+c x^2}}-\frac {35 c^2 (2 b B-3 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{8 b^{11/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 62, normalized size = 0.29 \begin {gather*} \frac {c^2 x^3 (2 b B-3 A c) \, _2F_1\left (-\frac {3}{2},3;-\frac {1}{2};\frac {c x}{b}+1\right )-A b^3}{3 b^4 x^{3/2} (x (b+c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.88, size = 166, normalized size = 0.78 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (-8 A b^4+18 A b^3 c x-63 A b^2 c^2 x^2-420 A b c^3 x^3-315 A c^4 x^4-12 b^4 B x+42 b^3 B c x^2+280 b^2 B c^2 x^3+210 b B c^3 x^4\right )}{24 b^5 x^{7/2} (b+c x)^2}-\frac {35 \left (2 b B c^2-3 A c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x+c x^2}}\right )}{8 b^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 477, normalized size = 2.23 \begin {gather*} \left [-\frac {105 \, {\left ({\left (2 \, B b c^{4} - 3 \, A c^{5}\right )} x^{6} + 2 \, {\left (2 \, B b^{2} c^{3} - 3 \, A b c^{4}\right )} x^{5} + {\left (2 \, B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} x^{4}\right )} \sqrt {b} \log \left (-\frac {c x^{2} + 2 \, b x + 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) + 2 \, {\left (8 \, A b^{5} - 105 \, {\left (2 \, B b^{2} c^{3} - 3 \, A b c^{4}\right )} x^{4} - 140 \, {\left (2 \, B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} x^{3} - 21 \, {\left (2 \, B b^{4} c - 3 \, A b^{3} c^{2}\right )} x^{2} + 6 \, {\left (2 \, B b^{5} - 3 \, A b^{4} c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{48 \, {\left (b^{6} c^{2} x^{6} + 2 \, b^{7} c x^{5} + b^{8} x^{4}\right )}}, \frac {105 \, {\left ({\left (2 \, B b c^{4} - 3 \, A c^{5}\right )} x^{6} + 2 \, {\left (2 \, B b^{2} c^{3} - 3 \, A b c^{4}\right )} x^{5} + {\left (2 \, B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} x^{4}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) - {\left (8 \, A b^{5} - 105 \, {\left (2 \, B b^{2} c^{3} - 3 \, A b c^{4}\right )} x^{4} - 140 \, {\left (2 \, B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} x^{3} - 21 \, {\left (2 \, B b^{4} c - 3 \, A b^{3} c^{2}\right )} x^{2} + 6 \, {\left (2 \, B b^{5} - 3 \, A b^{4} c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{24 \, {\left (b^{6} c^{2} x^{6} + 2 \, b^{7} c x^{5} + b^{8} x^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 200, normalized size = 0.93 \begin {gather*} \frac {35 \, {\left (2 \, B b c^{2} - 3 \, A c^{3}\right )} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{8 \, \sqrt {-b} b^{5}} + \frac {210 \, {\left (c x + b\right )}^{4} B b c^{2} - 560 \, {\left (c x + b\right )}^{3} B b^{2} c^{2} + 462 \, {\left (c x + b\right )}^{2} B b^{3} c^{2} - 96 \, {\left (c x + b\right )} B b^{4} c^{2} - 16 \, B b^{5} c^{2} - 315 \, {\left (c x + b\right )}^{4} A c^{3} + 840 \, {\left (c x + b\right )}^{3} A b c^{3} - 693 \, {\left (c x + b\right )}^{2} A b^{2} c^{3} + 144 \, {\left (c x + b\right )} A b^{3} c^{3} + 16 \, A b^{4} c^{3}}{24 \, {\left ({\left (c x + b\right )}^{\frac {3}{2}} - \sqrt {c x + b} b\right )}^{3} b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 234, normalized size = 1.09 \begin {gather*} \frac {\sqrt {\left (c x +b \right ) x}\, \left (315 \sqrt {c x +b}\, A \,c^{4} x^{4} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-210 \sqrt {c x +b}\, B b \,c^{3} x^{4} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-315 A \sqrt {b}\, c^{4} x^{4}+210 B \,b^{\frac {3}{2}} c^{3} x^{4}+315 \sqrt {c x +b}\, A b \,c^{3} x^{3} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-210 \sqrt {c x +b}\, B \,b^{2} c^{2} x^{3} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-420 A \,b^{\frac {3}{2}} c^{3} x^{3}+280 B \,b^{\frac {5}{2}} c^{2} x^{3}-63 A \,b^{\frac {5}{2}} c^{2} x^{2}+42 B \,b^{\frac {7}{2}} c \,x^{2}+18 A \,b^{\frac {7}{2}} c x -12 B \,b^{\frac {9}{2}} x -8 A \,b^{\frac {9}{2}}\right )}{24 \left (c x +b \right )^{2} b^{\frac {11}{2}} x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {B x + A}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}} x^{\frac {3}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{x^{3/2}\,{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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