Optimal. Leaf size=170 \[ -\frac {5 b^2 (7 b B-6 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{9/2}}+\frac {5 b \sqrt {b x+c x^2} (7 b B-6 A c)}{8 c^4}-\frac {5 x \sqrt {b x+c x^2} (7 b B-6 A c)}{12 c^3}+\frac {x^2 \sqrt {b x+c x^2} (7 b B-6 A c)}{3 b c^2}-\frac {2 x^4 (b B-A c)}{b c \sqrt {b x+c x^2}} \]
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Rubi [A] time = 0.15, antiderivative size = 170, 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 = {788, 670, 640, 620, 206} \begin {gather*} -\frac {5 b^2 (7 b B-6 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{9/2}}+\frac {x^2 \sqrt {b x+c x^2} (7 b B-6 A c)}{3 b c^2}-\frac {5 x \sqrt {b x+c x^2} (7 b B-6 A c)}{12 c^3}+\frac {5 b \sqrt {b x+c x^2} (7 b B-6 A c)}{8 c^4}-\frac {2 x^4 (b B-A c)}{b c \sqrt {b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 640
Rule 670
Rule 788
Rubi steps
\begin {align*} \int \frac {x^4 (A+B x)}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (b B-A c) x^4}{b c \sqrt {b x+c x^2}}-\left (\frac {6 A}{b}-\frac {7 B}{c}\right ) \int \frac {x^3}{\sqrt {b x+c x^2}} \, dx\\ &=-\frac {2 (b B-A c) x^4}{b c \sqrt {b x+c x^2}}+\frac {(7 b B-6 A c) x^2 \sqrt {b x+c x^2}}{3 b c^2}-\frac {(5 (7 b B-6 A c)) \int \frac {x^2}{\sqrt {b x+c x^2}} \, dx}{6 c^2}\\ &=-\frac {2 (b B-A c) x^4}{b c \sqrt {b x+c x^2}}-\frac {5 (7 b B-6 A c) x \sqrt {b x+c x^2}}{12 c^3}+\frac {(7 b B-6 A c) x^2 \sqrt {b x+c x^2}}{3 b c^2}+\frac {(5 b (7 b B-6 A c)) \int \frac {x}{\sqrt {b x+c x^2}} \, dx}{8 c^3}\\ &=-\frac {2 (b B-A c) x^4}{b c \sqrt {b x+c x^2}}+\frac {5 b (7 b B-6 A c) \sqrt {b x+c x^2}}{8 c^4}-\frac {5 (7 b B-6 A c) x \sqrt {b x+c x^2}}{12 c^3}+\frac {(7 b B-6 A c) x^2 \sqrt {b x+c x^2}}{3 b c^2}-\frac {\left (5 b^2 (7 b B-6 A c)\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{16 c^4}\\ &=-\frac {2 (b B-A c) x^4}{b c \sqrt {b x+c x^2}}+\frac {5 b (7 b B-6 A c) \sqrt {b x+c x^2}}{8 c^4}-\frac {5 (7 b B-6 A c) x \sqrt {b x+c x^2}}{12 c^3}+\frac {(7 b B-6 A c) x^2 \sqrt {b x+c x^2}}{3 b c^2}-\frac {\left (5 b^2 (7 b B-6 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{8 c^4}\\ &=-\frac {2 (b B-A c) x^4}{b c \sqrt {b x+c x^2}}+\frac {5 b (7 b B-6 A c) \sqrt {b x+c x^2}}{8 c^4}-\frac {5 (7 b B-6 A c) x \sqrt {b x+c x^2}}{12 c^3}+\frac {(7 b B-6 A c) x^2 \sqrt {b x+c x^2}}{3 b c^2}-\frac {5 b^2 (7 b B-6 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 137, normalized size = 0.81 \begin {gather*} \frac {\frac {(b+c x) (7 b B-6 A c) \left (c x \sqrt {\frac {c x}{b}+1} \left (15 b^2-10 b c x+8 c^2 x^2\right )-15 b^{5/2} \sqrt {c} \sqrt {x} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )\right )}{3 \sqrt {\frac {c x}{b}+1}}+16 c^4 x^4 (A c-b B)}{8 b c^5 \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.62, size = 136, normalized size = 0.80 \begin {gather*} \frac {5 \left (7 b^3 B-6 A b^2 c\right ) \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right )}{16 c^{9/2}}+\frac {\sqrt {b x+c x^2} \left (-90 A b^2 c-30 A b c^2 x+12 A c^3 x^2+105 b^3 B+35 b^2 B c x-14 b B c^2 x^2+8 B c^3 x^3\right )}{24 c^4 (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 313, normalized size = 1.84 \begin {gather*} \left [-\frac {15 \, {\left (7 \, B b^{4} - 6 \, A b^{3} c + {\left (7 \, B b^{3} c - 6 \, A b^{2} c^{2}\right )} x\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (8 \, B c^{4} x^{3} + 105 \, B b^{3} c - 90 \, A b^{2} c^{2} - 2 \, {\left (7 \, B b c^{3} - 6 \, A c^{4}\right )} x^{2} + 5 \, {\left (7 \, B b^{2} c^{2} - 6 \, A b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{48 \, {\left (c^{6} x + b c^{5}\right )}}, \frac {15 \, {\left (7 \, B b^{4} - 6 \, A b^{3} c + {\left (7 \, B b^{3} c - 6 \, A b^{2} c^{2}\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (8 \, B c^{4} x^{3} + 105 \, B b^{3} c - 90 \, A b^{2} c^{2} - 2 \, {\left (7 \, B b c^{3} - 6 \, A c^{4}\right )} x^{2} + 5 \, {\left (7 \, B b^{2} c^{2} - 6 \, A b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{24 \, {\left (c^{6} x + b c^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 163, normalized size = 0.96 \begin {gather*} \frac {1}{24} \, \sqrt {c x^{2} + b x} {\left (2 \, x {\left (\frac {4 \, B x}{c^{2}} - \frac {11 \, B b c^{10} - 6 \, A c^{11}}{c^{13}}\right )} + \frac {3 \, {\left (19 \, B b^{2} c^{9} - 14 \, A b c^{10}\right )}}{c^{13}}\right )} + \frac {5 \, {\left (7 \, B b^{3} - 6 \, A b^{2} c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{16 \, c^{\frac {9}{2}}} + \frac {2 \, {\left (B b^{4} - A b^{3} c\right )}}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} c + b \sqrt {c}\right )} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 215, normalized size = 1.26 \begin {gather*} \frac {B \,x^{4}}{3 \sqrt {c \,x^{2}+b x}\, c}+\frac {A \,x^{3}}{2 \sqrt {c \,x^{2}+b x}\, c}-\frac {7 B b \,x^{3}}{12 \sqrt {c \,x^{2}+b x}\, c^{2}}-\frac {5 A b \,x^{2}}{4 \sqrt {c \,x^{2}+b x}\, c^{2}}+\frac {35 B \,b^{2} x^{2}}{24 \sqrt {c \,x^{2}+b x}\, c^{3}}-\frac {15 A \,b^{2} x}{4 \sqrt {c \,x^{2}+b x}\, c^{3}}+\frac {35 B \,b^{3} x}{8 \sqrt {c \,x^{2}+b x}\, c^{4}}+\frac {15 A \,b^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {7}{2}}}-\frac {35 B \,b^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{16 c^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.94, size = 212, normalized size = 1.25 \begin {gather*} \frac {B x^{4}}{3 \, \sqrt {c x^{2} + b x} c} - \frac {7 \, B b x^{3}}{12 \, \sqrt {c x^{2} + b x} c^{2}} + \frac {A x^{3}}{2 \, \sqrt {c x^{2} + b x} c} + \frac {35 \, B b^{2} x^{2}}{24 \, \sqrt {c x^{2} + b x} c^{3}} - \frac {5 \, A b x^{2}}{4 \, \sqrt {c x^{2} + b x} c^{2}} + \frac {35 \, B b^{3} x}{8 \, \sqrt {c x^{2} + b x} c^{4}} - \frac {15 \, A b^{2} x}{4 \, \sqrt {c x^{2} + b x} c^{3}} - \frac {35 \, B b^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{16 \, c^{\frac {9}{2}}} + \frac {15 \, A b^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, c^{\frac {7}{2}}} \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 = -0.01 \begin {gather*} \int \frac {x^4\,\left (A+B\,x\right )}{{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \left (A + B x\right )}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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