Optimal. Leaf size=84 \[ -\frac {2 x^3 (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {2 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{5/2}}-\frac {2 B x}{c^2 \sqrt {b x+c x^2}} \]
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Rubi [A] time = 0.08, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {788, 652, 620, 206} \begin {gather*} -\frac {2 x^3 (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}}-\frac {2 B x}{c^2 \sqrt {b x+c x^2}}+\frac {2 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 652
Rule 788
Rubi steps
\begin {align*} \int \frac {x^3 (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (b B-A c) x^3}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {B \int \frac {x^2}{\left (b x+c x^2\right )^{3/2}} \, dx}{c}\\ &=-\frac {2 (b B-A c) x^3}{3 b c \left (b x+c x^2\right )^{3/2}}-\frac {2 B x}{c^2 \sqrt {b x+c x^2}}+\frac {B \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{c^2}\\ &=-\frac {2 (b B-A c) x^3}{3 b c \left (b x+c x^2\right )^{3/2}}-\frac {2 B x}{c^2 \sqrt {b x+c x^2}}+\frac {(2 B) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{c^2}\\ &=-\frac {2 (b B-A c) x^3}{3 b c \left (b x+c x^2\right )^{3/2}}-\frac {2 B x}{c^2 \sqrt {b x+c x^2}}+\frac {2 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 99, normalized size = 1.18 \begin {gather*} \frac {x \left (2 \sqrt {c} x \left (A c^2 x-3 b^2 B-4 b B c x\right )+6 b^{3/2} B \sqrt {x} (b+c x) \sqrt {\frac {c x}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )\right )}{3 b c^{5/2} (x (b+c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.44, size = 92, normalized size = 1.10 \begin {gather*} -\frac {2 \sqrt {b x+c x^2} \left (-A c^2 x+3 b^2 B+4 b B c x\right )}{3 b c^2 (b+c x)^2}-\frac {B \log \left (-2 c^{5/2} \sqrt {b x+c x^2}+b c^2+2 c^3 x\right )}{c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 239, normalized size = 2.85 \begin {gather*} \left [\frac {3 \, {\left (B b c^{2} x^{2} + 2 \, B b^{2} c x + B b^{3}\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (3 \, B b^{2} c + {\left (4 \, B b c^{2} - A c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (b c^{5} x^{2} + 2 \, b^{2} c^{4} x + b^{3} c^{3}\right )}}, -\frac {2 \, {\left (3 \, {\left (B b c^{2} x^{2} + 2 \, B b^{2} c x + B b^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (3 \, B b^{2} c + {\left (4 \, B b c^{2} - A c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}\right )}}{3 \, {\left (b c^{5} x^{2} + 2 \, b^{2} c^{4} x + b^{3} c^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 206, normalized size = 2.45 \begin {gather*} -\frac {B \,x^{3}}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c}-\frac {A \,x^{2}}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} c}+\frac {B b \,x^{2}}{2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{2}}-\frac {A b x}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{2}}+\frac {B \,b^{2} x}{6 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{3}}+\frac {2 A x}{3 \sqrt {c \,x^{2}+b x}\, b c}-\frac {7 B x}{3 \sqrt {c \,x^{2}+b x}\, c^{2}}+\frac {B \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{c^{\frac {5}{2}}}+\frac {A}{3 \sqrt {c \,x^{2}+b x}\, c^{2}}-\frac {B b}{6 \sqrt {c \,x^{2}+b x}\, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.67, size = 221, normalized size = 2.63 \begin {gather*} -\frac {1}{3} \, B x {\left (\frac {3 \, x^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {b x}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} - \frac {2 \, x}{\sqrt {c x^{2} + b x} b c} - \frac {1}{\sqrt {c x^{2} + b x} c^{2}}\right )} - \frac {A x^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} - \frac {4 \, B x}{3 \, \sqrt {c x^{2} + b x} c^{2}} - \frac {A b x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} + \frac {2 \, A x}{3 \, \sqrt {c x^{2} + b x} b c} + \frac {B \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{c^{\frac {5}{2}}} + \frac {A}{3 \, \sqrt {c x^{2} + b x} c^{2}} - \frac {2 \, \sqrt {c x^{2} + b x} B}{3 \, b c^{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^3\,\left (A+B\,x\right )}{{\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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \left (A + B x\right )}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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