Optimal. Leaf size=63 \[ \frac {(a+b x)^m \left (a^2-b^2 x^2\right )^{1+p} \, _2F_1\left (1,2+m+2 p;2+m+p;\frac {a+b x}{2 a}\right )}{2 a b (1+m+p)} \]
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Rubi [A]
time = 0.04, antiderivative size = 85, normalized size of antiderivative = 1.35, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {694, 692, 71}
\begin {gather*} -\frac {2^{m+p} (a+b x)^m \left (a^2-b^2 x^2\right )^{p+1} \left (\frac {b x}{a}+1\right )^{-m-p-1} \, _2F_1\left (-m-p,p+1;p+2;\frac {a-b x}{2 a}\right )}{a b (p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 692
Rule 694
Rubi steps
\begin {align*} \int (a+b x)^m \left (a^2-b^2 x^2\right )^p \, dx &=\left ((a+b x)^m \left (1+\frac {b x}{a}\right )^{-m}\right ) \int \left (1+\frac {b x}{a}\right )^m \left (a^2-b^2 x^2\right )^p \, dx\\ &=\left ((a+b x)^m \left (1+\frac {b x}{a}\right )^{-1-m-p} \left (a^2-a b x\right )^{-1-p} \left (a^2-b^2 x^2\right )^{1+p}\right ) \int \left (1+\frac {b x}{a}\right )^{m+p} \left (a^2-a b x\right )^p \, dx\\ &=-\frac {2^{m+p} (a+b x)^m \left (1+\frac {b x}{a}\right )^{-1-m-p} \left (a^2-b^2 x^2\right )^{1+p} \, _2F_1\left (-m-p,1+p;2+p;\frac {a-b x}{2 a}\right )}{a b (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 85, normalized size = 1.35 \begin {gather*} \frac {2^{m+p} (-a+b x) (a+b x)^m \left (1+\frac {b x}{a}\right )^{-m-p} \left (a^2-b^2 x^2\right )^p \, _2F_1\left (-m-p,1+p;2+p;\frac {a-b x}{2 a}\right )}{b (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int \left (b x +a \right )^{m} \left (-b^{2} x^{2}+a^{2}\right )^{p}\, dx\]
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 (- \left (- a + b x\right ) \left (a + b x\right )\right )^{p} \left (a + b x\right )^{m}\, 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 (a^2-b^2\,x^2\right )}^p\,{\left (a+b\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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