Optimal. Leaf size=71 \[ -\frac {2 b \left (-\frac {c x}{b}\right )^{-\frac {1}{2}-m} (d x)^m (b+c x) \left (b x+c x^2\right )^{3/2} \, _2F_1\left (\frac {5}{2},-\frac {3}{2}-m;\frac {7}{2};1+\frac {c x}{b}\right )}{5 c^2 x} \]
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Rubi [A]
time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {688, 69, 67}
\begin {gather*} -\frac {2 b (b+c x) \left (b x+c x^2\right )^{3/2} (d x)^m \left (-\frac {c x}{b}\right )^{-m-\frac {1}{2}} \, _2F_1\left (\frac {5}{2},-m-\frac {3}{2};\frac {7}{2};\frac {c x}{b}+1\right )}{5 c^2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 69
Rule 688
Rubi steps
\begin {align*} \int (d x)^m \left (b x+c x^2\right )^{3/2} \, dx &=\frac {\left (x^{-\frac {3}{2}-m} (d x)^m \left (b x+c x^2\right )^{3/2}\right ) \int x^{\frac {3}{2}+m} (b+c x)^{3/2} \, dx}{(b+c x)^{3/2}}\\ &=-\frac {\left (b \left (-\frac {c x}{b}\right )^{-\frac {1}{2}-m} (d x)^m \left (b x+c x^2\right )^{3/2}\right ) \int \left (-\frac {c x}{b}\right )^{\frac {3}{2}+m} (b+c x)^{3/2} \, dx}{c x (b+c x)^{3/2}}\\ &=-\frac {2 b \left (-\frac {c x}{b}\right )^{-\frac {1}{2}-m} (d x)^m (b+c x) \left (b x+c x^2\right )^{3/2} \, _2F_1\left (\frac {5}{2},-\frac {3}{2}-m;\frac {7}{2};1+\frac {c x}{b}\right )}{5 c^2 x}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 60, normalized size = 0.85 \begin {gather*} -\frac {2 \left (-\frac {c x}{b}\right )^{-\frac {5}{2}-m} (d x)^m (x (b+c x))^{5/2} \, _2F_1\left (\frac {5}{2},-\frac {3}{2}-m;\frac {7}{2};1+\frac {c x}{b}\right )}{5 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \left (d x \right )^{m} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 \left (d x\right )^{m} \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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {\left (c\,x^2+b\,x\right )}^{3/2}\,{\left (d\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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