Optimal. Leaf size=136 \[ -\frac {(5 b B-2 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{7/2}}+\frac {\sqrt {b x+c x^2} (5 b B-2 A c)}{b c^3}-\frac {2 x^2 (5 b B-2 A c)}{3 b c^2 \sqrt {b x+c x^2}}-\frac {2 x^4 (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {788, 668, 640, 620, 206} \begin {gather*} -\frac {2 x^2 (5 b B-2 A c)}{3 b c^2 \sqrt {b x+c x^2}}+\frac {\sqrt {b x+c x^2} (5 b B-2 A c)}{b c^3}-\frac {(5 b B-2 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{7/2}}-\frac {2 x^4 (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 640
Rule 668
Rule 788
Rubi steps
\begin {align*} \int \frac {x^4 (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (b B-A c) x^4}{3 b c \left (b x+c x^2\right )^{3/2}}-\frac {1}{3} \left (\frac {2 A}{b}-\frac {5 B}{c}\right ) \int \frac {x^3}{\left (b x+c x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (b B-A c) x^4}{3 b c \left (b x+c x^2\right )^{3/2}}-\frac {2 (5 b B-2 A c) x^2}{3 b c^2 \sqrt {b x+c x^2}}+\frac {(5 b B-2 A c) \int \frac {x}{\sqrt {b x+c x^2}} \, dx}{b c^2}\\ &=-\frac {2 (b B-A c) x^4}{3 b c \left (b x+c x^2\right )^{3/2}}-\frac {2 (5 b B-2 A c) x^2}{3 b c^2 \sqrt {b x+c x^2}}+\frac {(5 b B-2 A c) \sqrt {b x+c x^2}}{b c^3}-\frac {(5 b B-2 A c) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{2 c^3}\\ &=-\frac {2 (b B-A c) x^4}{3 b c \left (b x+c x^2\right )^{3/2}}-\frac {2 (5 b B-2 A c) x^2}{3 b c^2 \sqrt {b x+c x^2}}+\frac {(5 b B-2 A c) \sqrt {b x+c x^2}}{b c^3}-\frac {(5 b B-2 A c) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{c^3}\\ &=-\frac {2 (b B-A c) x^4}{3 b c \left (b x+c x^2\right )^{3/2}}-\frac {2 (5 b B-2 A c) x^2}{3 b c^2 \sqrt {b x+c x^2}}+\frac {(5 b B-2 A c) \sqrt {b x+c x^2}}{b c^3}-\frac {(5 b B-2 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 80, normalized size = 0.59 \begin {gather*} \frac {2 x^4 \left ((b+c x) \sqrt {\frac {c x}{b}+1} (5 b B-2 A c) \, _2F_1\left (\frac {3}{2},\frac {5}{2};\frac {7}{2};-\frac {c x}{b}\right )+5 b (A c-b B)\right )}{15 b^2 c (x (b+c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.51, size = 107, normalized size = 0.79 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (-6 A b c-8 A c^2 x+15 b^2 B+20 b B c x+3 B c^2 x^2\right )}{3 c^3 (b+c x)^2}+\frac {(5 b B-2 A c) \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right )}{2 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 321, normalized size = 2.36 \begin {gather*} \left [-\frac {3 \, {\left (5 \, B b^{3} - 2 \, A b^{2} c + {\left (5 \, B b c^{2} - 2 \, A c^{3}\right )} x^{2} + 2 \, {\left (5 \, B b^{2} c - 2 \, A b 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 (3 \, B c^{3} x^{2} + 15 \, B b^{2} c - 6 \, A b c^{2} + 4 \, {\left (5 \, B b c^{2} - 2 \, A c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{6 \, {\left (c^{6} x^{2} + 2 \, b c^{5} x + b^{2} c^{4}\right )}}, \frac {3 \, {\left (5 \, B b^{3} - 2 \, A b^{2} c + {\left (5 \, B b c^{2} - 2 \, A c^{3}\right )} x^{2} + 2 \, {\left (5 \, B b^{2} c - 2 \, A b c^{2}\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (3 \, B c^{3} x^{2} + 15 \, B b^{2} c - 6 \, A b c^{2} + 4 \, {\left (5 \, B b c^{2} - 2 \, A c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (c^{6} x^{2} + 2 \, b c^{5} x + b^{2} c^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 224, normalized size = 1.65 \begin {gather*} \frac {\sqrt {c x^{2} + b x} B}{c^{3}} + \frac {{\left (5 \, B b - 2 \, A c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{2 \, c^{\frac {7}{2}}} + \frac {2 \, {\left (9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{2} c^{\frac {3}{2}} - 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b c^{\frac {5}{2}} + 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} B b^{3} c - 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{2} c^{2} + 7 \, B b^{4} \sqrt {c} - 4 \, A b^{3} c^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b\right )}^{3} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 283, normalized size = 2.08 \begin {gather*} \frac {B \,x^{4}}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} c}-\frac {A \,x^{3}}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c}+\frac {5 B b \,x^{3}}{6 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{2}}+\frac {A b \,x^{2}}{2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{2}}-\frac {5 B \,b^{2} x^{2}}{4 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{3}}+\frac {A \,b^{2} x}{6 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{3}}-\frac {5 B \,b^{3} x}{12 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{4}}-\frac {7 A x}{3 \sqrt {c \,x^{2}+b x}\, c^{2}}+\frac {35 B b x}{6 \sqrt {c \,x^{2}+b x}\, c^{3}}+\frac {A \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{c^{\frac {5}{2}}}-\frac {5 B b \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {7}{2}}}-\frac {A b}{6 \sqrt {c \,x^{2}+b x}\, c^{3}}+\frac {5 B \,b^{2}}{12 \sqrt {c \,x^{2}+b x}\, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.60, size = 310, normalized size = 2.28 \begin {gather*} -\frac {1}{3} \, A 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 {5 \, B 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 )}}{6 \, c} + \frac {B x^{4}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {10 \, B b x}{3 \, \sqrt {c x^{2} + b x} c^{3}} - \frac {4 \, A x}{3 \, \sqrt {c x^{2} + b x} c^{2}} - \frac {5 \, B b \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2 \, c^{\frac {7}{2}}} + \frac {A \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{c^{\frac {5}{2}}} + \frac {5 \, \sqrt {c x^{2} + b x} B}{3 \, c^{3}} - \frac {2 \, \sqrt {c x^{2} + b x} A}{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^4\,\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^{4} \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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