Optimal. Leaf size=80 \[ -\frac {16 b^2 \sqrt {x}}{3 c^3 \sqrt {b x+c x^2}}-\frac {8 b x^{3/2}}{3 c^2 \sqrt {b x+c x^2}}+\frac {2 x^{5/2}}{3 c \sqrt {b x+c x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {656, 648} \begin {gather*} -\frac {16 b^2 \sqrt {x}}{3 c^3 \sqrt {b x+c x^2}}-\frac {8 b x^{3/2}}{3 c^2 \sqrt {b x+c x^2}}+\frac {2 x^{5/2}}{3 c \sqrt {b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rubi steps
\begin {align*} \int \frac {x^{7/2}}{\left (b x+c x^2\right )^{3/2}} \, dx &=\frac {2 x^{5/2}}{3 c \sqrt {b x+c x^2}}-\frac {(4 b) \int \frac {x^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 c}\\ &=-\frac {8 b x^{3/2}}{3 c^2 \sqrt {b x+c x^2}}+\frac {2 x^{5/2}}{3 c \sqrt {b x+c x^2}}+\frac {\left (8 b^2\right ) \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 c^2}\\ &=-\frac {16 b^2 \sqrt {x}}{3 c^3 \sqrt {b x+c x^2}}-\frac {8 b x^{3/2}}{3 c^2 \sqrt {b x+c x^2}}+\frac {2 x^{5/2}}{3 c \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 41, normalized size = 0.51 \begin {gather*} \frac {2 \sqrt {x} \left (-8 b^2-4 b c x+c^2 x^2\right )}{3 c^3 \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.48, size = 50, normalized size = 0.62 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (-8 b^2-4 b c x+c^2 x^2\right )}{3 c^3 \sqrt {x} (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 50, normalized size = 0.62 \begin {gather*} \frac {2 \, {\left (c^{2} x^{2} - 4 \, b c x - 8 \, b^{2}\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{3 \, {\left (c^{4} x^{2} + b c^{3} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 54, normalized size = 0.68 \begin {gather*} \frac {16 \, b^{\frac {3}{2}}}{3 \, c^{3}} - \frac {2 \, b^{2}}{\sqrt {c x + b} c^{3}} + \frac {2 \, {\left ({\left (c x + b\right )}^{\frac {3}{2}} c^{6} - 6 \, \sqrt {c x + b} b c^{6}\right )}}{3 \, c^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 44, normalized size = 0.55 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (-c^{2} x^{2}+4 b c x +8 b^{2}\right ) x^{\frac {3}{2}}}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {2 \, {\left ({\left (c^{3} x^{2} - b c^{2} x - 2 \, b^{2} c\right )} x^{2} - 2 \, {\left (b c^{2} x^{2} + 2 \, b^{2} c x + b^{3}\right )} x\right )}}{3 \, {\left (c^{4} x^{2} + b c^{3} x\right )} \sqrt {c x + b}} + \int \frac {2 \, {\left (b^{2} c x + b^{3}\right )} x}{{\left (c^{4} x^{3} + 2 \, b c^{3} x^{2} + b^{2} c^{2} x\right )} \sqrt {c x + b}}\,{d x} \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^{7/2}}{{\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 \frac {x^{\frac {7}{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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