Optimal. Leaf size=64 \[ \frac {\sqrt {c x^2} (d x)^{m+2} (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \, _2F_1\left (m+2,-n;m+3;-\frac {b x}{a}\right )}{d^2 (m+2) x} \]
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Rubi [A] time = 0.03, antiderivative size = 64, 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, 66, 64} \[ \frac {\sqrt {c x^2} (d x)^{m+2} (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \, _2F_1\left (m+2,-n;m+3;-\frac {b x}{a}\right )}{d^2 (m+2) x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 16
Rule 64
Rule 66
Rubi steps
\begin {align*} \int (d x)^m \sqrt {c x^2} (a+b x)^n \, dx &=\frac {\sqrt {c x^2} \int x (d x)^m (a+b x)^n \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int (d x)^{1+m} (a+b x)^n \, dx}{d x}\\ &=\frac {\left (\sqrt {c x^2} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n}\right ) \int (d x)^{1+m} \left (1+\frac {b x}{a}\right )^n \, dx}{d x}\\ &=\frac {(d x)^{2+m} \sqrt {c x^2} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (2+m,-n;3+m;-\frac {b x}{a}\right )}{d^2 (2+m) x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 57, normalized size = 0.89 \[ \frac {x \sqrt {c x^2} (d x)^m (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \, _2F_1\left (m+2,-n;m+3;-\frac {b x}{a}\right )}{m+2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c x^{2}} {\left (b x + a\right )}^{n} \left (d x\right )^{m}, x\right ) \]
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 \[ \int \sqrt {c x^{2}} {\left (b x + a\right )}^{n} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \sqrt {c \,x^{2}}\, \left (d x \right )^{m} \left (b x +a \right )^{n}\, dx \]
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 \[ \int \sqrt {c x^{2}} {\left (b x + a\right )}^{n} \left (d x\right )^{m}\,{d x} \]
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 \[ \int {\left (d\,x\right )}^m\,\sqrt {c\,x^2}\,{\left (a+b\,x\right )}^n \,d x \]
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 \[ \int \sqrt {c x^{2}} \left (d x\right )^{m} \left (a + b x\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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