Optimal. Leaf size=65 \[ \frac {c (d x)^{4+m} \sqrt {c x^2} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (4+m,-n;5+m;-\frac {b x}{a}\right )}{d^4 (4+m) x} \]
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Rubi [A]
time = 0.02, antiderivative size = 65, 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 = {15, 16, 68, 66}
\begin {gather*} \frac {c \sqrt {c x^2} (d x)^{m+4} (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \, _2F_1\left (m+4,-n;m+5;-\frac {b x}{a}\right )}{d^4 (m+4) x} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 16
Rule 66
Rule 68
Rubi steps
\begin {align*} \int (d x)^m \left (c x^2\right )^{3/2} (a+b x)^n \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int x^3 (d x)^m (a+b x)^n \, dx}{x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int (d x)^{3+m} (a+b x)^n \, dx}{d^3 x}\\ &=\frac {\left (c \sqrt {c x^2} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n}\right ) \int (d x)^{3+m} \left (1+\frac {b x}{a}\right )^n \, dx}{d^3 x}\\ &=\frac {c (d x)^{4+m} \sqrt {c x^2} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (4+m,-n;5+m;-\frac {b x}{a}\right )}{d^4 (4+m) x}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 57, normalized size = 0.88 \begin {gather*} \frac {x (d x)^m \left (c x^2\right )^{3/2} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (4+m,-n;5+m;-\frac {b x}{a}\right )}{4+m} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (d x \right )^{m} \left (c \,x^{2}\right )^{\frac {3}{2}} \left (b x +a \right )^{n}\, dx\]
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 (c x^{2}\right )^{\frac {3}{2}} \left (d x\right )^{m} \left (a + b x\right )^{n}\, 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 = -0.02 \begin {gather*} \int {\left (d\,x\right )}^m\,{\left (c\,x^2\right )}^{3/2}\,{\left (a+b\,x\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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