Optimal. Leaf size=134 \[ \frac {7 b^6 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{512 c^{9/2}}-\frac {7 b^4 (b+2 c x) \sqrt {b x+c x^2}}{512 c^4}+\frac {7 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac {7 b \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac {x \left (b x+c x^2\right )^{5/2}}{6 c} \]
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Rubi [A] time = 0.06, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {670, 640, 612, 620, 206} \begin {gather*} -\frac {7 b^4 (b+2 c x) \sqrt {b x+c x^2}}{512 c^4}+\frac {7 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 b^6 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{512 c^{9/2}}-\frac {7 b \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac {x \left (b x+c x^2\right )^{5/2}}{6 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 620
Rule 640
Rule 670
Rubi steps
\begin {align*} \int x^2 \left (b x+c x^2\right )^{3/2} \, dx &=\frac {x \left (b x+c x^2\right )^{5/2}}{6 c}-\frac {(7 b) \int x \left (b x+c x^2\right )^{3/2} \, dx}{12 c}\\ &=-\frac {7 b \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac {x \left (b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (7 b^2\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{24 c^2}\\ &=\frac {7 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac {7 b \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac {x \left (b x+c x^2\right )^{5/2}}{6 c}-\frac {\left (7 b^4\right ) \int \sqrt {b x+c x^2} \, dx}{128 c^3}\\ &=-\frac {7 b^4 (b+2 c x) \sqrt {b x+c x^2}}{512 c^4}+\frac {7 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac {7 b \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac {x \left (b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (7 b^6\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{1024 c^4}\\ &=-\frac {7 b^4 (b+2 c x) \sqrt {b x+c x^2}}{512 c^4}+\frac {7 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac {7 b \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac {x \left (b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (7 b^6\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{512 c^4}\\ &=-\frac {7 b^4 (b+2 c x) \sqrt {b x+c x^2}}{512 c^4}+\frac {7 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac {7 b \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac {x \left (b x+c x^2\right )^{5/2}}{6 c}+\frac {7 b^6 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{512 c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 120, normalized size = 0.90 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\frac {105 b^{11/2} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}+\sqrt {c} \left (-105 b^5+70 b^4 c x-56 b^3 c^2 x^2+48 b^2 c^3 x^3+1664 b c^4 x^4+1280 c^5 x^5\right )\right )}{7680 c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.35, size = 112, normalized size = 0.84 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (-105 b^5+70 b^4 c x-56 b^3 c^2 x^2+48 b^2 c^3 x^3+1664 b c^4 x^4+1280 c^5 x^5\right )}{7680 c^4}-\frac {7 b^6 \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right )}{1024 c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 214, normalized size = 1.60 \begin {gather*} \left [\frac {105 \, b^{6} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (1280 \, c^{6} x^{5} + 1664 \, b c^{5} x^{4} + 48 \, b^{2} c^{4} x^{3} - 56 \, b^{3} c^{3} x^{2} + 70 \, b^{4} c^{2} x - 105 \, b^{5} c\right )} \sqrt {c x^{2} + b x}}{15360 \, c^{5}}, -\frac {105 \, b^{6} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (1280 \, c^{6} x^{5} + 1664 \, b c^{5} x^{4} + 48 \, b^{2} c^{4} x^{3} - 56 \, b^{3} c^{3} x^{2} + 70 \, b^{4} c^{2} x - 105 \, b^{5} c\right )} \sqrt {c x^{2} + b x}}{7680 \, c^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 108, normalized size = 0.81 \begin {gather*} -\frac {7 \, b^{6} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{1024 \, c^{\frac {9}{2}}} + \frac {1}{7680} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c x + 13 \, b\right )} x + \frac {3 \, b^{2}}{c}\right )} x - \frac {7 \, b^{3}}{c^{2}}\right )} x + \frac {35 \, b^{4}}{c^{3}}\right )} x - \frac {105 \, b^{5}}{c^{4}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 146, normalized size = 1.09 \begin {gather*} \frac {7 b^{6} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{1024 c^{\frac {9}{2}}}-\frac {7 \sqrt {c \,x^{2}+b x}\, b^{4} x}{256 c^{3}}-\frac {7 \sqrt {c \,x^{2}+b x}\, b^{5}}{512 c^{4}}+\frac {7 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{2} x}{96 c^{2}}+\frac {7 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{3}}{192 c^{3}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} x}{6 c}-\frac {7 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} b}{60 c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 144, normalized size = 1.07 \begin {gather*} -\frac {7 \, \sqrt {c x^{2} + b x} b^{4} x}{256 \, c^{3}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2} x}{96 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} x}{6 \, c} + \frac {7 \, b^{6} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {9}{2}}} - \frac {7 \, \sqrt {c x^{2} + b x} b^{5}}{512 \, c^{4}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3}}{192 \, c^{3}} - \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b}{60 \, 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 x^2\,{\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 x^{2} \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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