Optimal. Leaf size=162 \[ \frac {5 b^3 (7 b B-8 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{9/2}}-\frac {5 b^2 \sqrt {b x+c x^2} (7 b B-8 A c)}{64 c^4}+\frac {5 b x \sqrt {b x+c x^2} (7 b B-8 A c)}{96 c^3}-\frac {x^2 \sqrt {b x+c x^2} (7 b B-8 A c)}{24 c^2}+\frac {B x^3 \sqrt {b x+c x^2}}{4 c} \]
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Rubi [A] time = 0.16, antiderivative size = 162, 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 = {794, 670, 640, 620, 206} \begin {gather*} -\frac {5 b^2 \sqrt {b x+c x^2} (7 b B-8 A c)}{64 c^4}+\frac {5 b^3 (7 b B-8 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{9/2}}+\frac {5 b x \sqrt {b x+c x^2} (7 b B-8 A c)}{96 c^3}-\frac {x^2 \sqrt {b x+c x^2} (7 b B-8 A c)}{24 c^2}+\frac {B x^3 \sqrt {b x+c x^2}}{4 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 640
Rule 670
Rule 794
Rubi steps
\begin {align*} \int \frac {x^3 (A+B x)}{\sqrt {b x+c x^2}} \, dx &=\frac {B x^3 \sqrt {b x+c x^2}}{4 c}+\frac {\left (3 (-b B+A c)+\frac {1}{2} (-b B+2 A c)\right ) \int \frac {x^3}{\sqrt {b x+c x^2}} \, dx}{4 c}\\ &=-\frac {(7 b B-8 A c) x^2 \sqrt {b x+c x^2}}{24 c^2}+\frac {B x^3 \sqrt {b x+c x^2}}{4 c}+\frac {(5 b (7 b B-8 A c)) \int \frac {x^2}{\sqrt {b x+c x^2}} \, dx}{48 c^2}\\ &=\frac {5 b (7 b B-8 A c) x \sqrt {b x+c x^2}}{96 c^3}-\frac {(7 b B-8 A c) x^2 \sqrt {b x+c x^2}}{24 c^2}+\frac {B x^3 \sqrt {b x+c x^2}}{4 c}-\frac {\left (5 b^2 (7 b B-8 A c)\right ) \int \frac {x}{\sqrt {b x+c x^2}} \, dx}{64 c^3}\\ &=-\frac {5 b^2 (7 b B-8 A c) \sqrt {b x+c x^2}}{64 c^4}+\frac {5 b (7 b B-8 A c) x \sqrt {b x+c x^2}}{96 c^3}-\frac {(7 b B-8 A c) x^2 \sqrt {b x+c x^2}}{24 c^2}+\frac {B x^3 \sqrt {b x+c x^2}}{4 c}+\frac {\left (5 b^3 (7 b B-8 A c)\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{128 c^4}\\ &=-\frac {5 b^2 (7 b B-8 A c) \sqrt {b x+c x^2}}{64 c^4}+\frac {5 b (7 b B-8 A c) x \sqrt {b x+c x^2}}{96 c^3}-\frac {(7 b B-8 A c) x^2 \sqrt {b x+c x^2}}{24 c^2}+\frac {B x^3 \sqrt {b x+c x^2}}{4 c}+\frac {\left (5 b^3 (7 b B-8 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{64 c^4}\\ &=-\frac {5 b^2 (7 b B-8 A c) \sqrt {b x+c x^2}}{64 c^4}+\frac {5 b (7 b B-8 A c) x \sqrt {b x+c x^2}}{96 c^3}-\frac {(7 b B-8 A c) x^2 \sqrt {b x+c x^2}}{24 c^2}+\frac {B x^3 \sqrt {b x+c x^2}}{4 c}+\frac {5 b^3 (7 b B-8 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 129, normalized size = 0.80 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\frac {15 b^{5/2} (7 b B-8 A c) \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}+\sqrt {c} \left (10 b^2 c (12 A+7 B x)-8 b c^2 x (10 A+7 B x)+16 c^3 x^2 (4 A+3 B x)-105 b^3 B\right )\right )}{192 c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.52, size = 129, normalized size = 0.80 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (120 A b^2 c-80 A b c^2 x+64 A c^3 x^2-105 b^3 B+70 b^2 B c x-56 b B c^2 x^2+48 B c^3 x^3\right )}{192 c^4}-\frac {5 \left (7 b^4 B-8 A b^3 c\right ) \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right )}{128 c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 256, normalized size = 1.58 \begin {gather*} \left [-\frac {15 \, {\left (7 \, B b^{4} - 8 \, A b^{3} c\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (48 \, B c^{4} x^{3} - 105 \, B b^{3} c + 120 \, A b^{2} c^{2} - 8 \, {\left (7 \, B b c^{3} - 8 \, A c^{4}\right )} x^{2} + 10 \, {\left (7 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{384 \, c^{5}}, -\frac {15 \, {\left (7 \, B b^{4} - 8 \, A b^{3} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (48 \, B c^{4} x^{3} - 105 \, B b^{3} c + 120 \, A b^{2} c^{2} - 8 \, {\left (7 \, B b c^{3} - 8 \, A c^{4}\right )} x^{2} + 10 \, {\left (7 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{192 \, c^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 137, normalized size = 0.85 \begin {gather*} \frac {1}{192} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (\frac {6 \, B x}{c} - \frac {7 \, B b c^{2} - 8 \, A c^{3}}{c^{4}}\right )} x + \frac {5 \, {\left (7 \, B b^{2} c - 8 \, A b c^{2}\right )}}{c^{4}}\right )} x - \frac {15 \, {\left (7 \, B b^{3} - 8 \, A b^{2} c\right )}}{c^{4}}\right )} - \frac {5 \, {\left (7 \, B b^{4} - 8 \, A b^{3} c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{128 \, c^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 209, normalized size = 1.29 \begin {gather*} \frac {\sqrt {c \,x^{2}+b x}\, B \,x^{3}}{4 c}+\frac {\sqrt {c \,x^{2}+b x}\, A \,x^{2}}{3 c}-\frac {7 \sqrt {c \,x^{2}+b x}\, B b \,x^{2}}{24 c^{2}}-\frac {5 A \,b^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{16 c^{\frac {7}{2}}}+\frac {35 B \,b^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{128 c^{\frac {9}{2}}}-\frac {5 \sqrt {c \,x^{2}+b x}\, A b x}{12 c^{2}}+\frac {35 \sqrt {c \,x^{2}+b x}\, B \,b^{2} x}{96 c^{3}}+\frac {5 \sqrt {c \,x^{2}+b x}\, A \,b^{2}}{8 c^{3}}-\frac {35 \sqrt {c \,x^{2}+b x}\, B \,b^{3}}{64 c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 206, normalized size = 1.27 \begin {gather*} \frac {\sqrt {c x^{2} + b x} B x^{3}}{4 \, c} - \frac {7 \, \sqrt {c x^{2} + b x} B b x^{2}}{24 \, c^{2}} + \frac {\sqrt {c x^{2} + b x} A x^{2}}{3 \, c} + \frac {35 \, \sqrt {c x^{2} + b x} B b^{2} x}{96 \, c^{3}} - \frac {5 \, \sqrt {c x^{2} + b x} A b x}{12 \, c^{2}} + \frac {35 \, B b^{4} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {9}{2}}} - \frac {5 \, A b^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{16 \, c^{\frac {7}{2}}} - \frac {35 \, \sqrt {c x^{2} + b x} B b^{3}}{64 \, c^{4}} + \frac {5 \, \sqrt {c x^{2} + b x} A b^{2}}{8 \, c^{3}} \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^3\,\left (A+B\,x\right )}{\sqrt {c\,x^2+b\,x}} \,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^{3} \left (A + B x\right )}{\sqrt {x \left (b + c x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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