Optimal. Leaf size=203 \[ \frac {x^2 (a+b x)^{1+n}}{b d (2+n) (c+d x)}-\frac {(a+b x)^{1+n} (c (b c (2+n) (a d+b c (3+n))-a d (a d+b c (5+3 n)))+d (b c-a d) (a d+b c (3+n)) x)}{b^2 d^3 (b c-a d) (1+n) (2+n) (c+d x)}-\frac {c^2 (3 a d-b c (3+n)) (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;-\frac {d (a+b x)}{b c-a d}\right )}{d^3 (b c-a d)^2 (1+n)} \]
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Rubi [A]
time = 0.14, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {102, 151, 70}
\begin {gather*} -\frac {(a+b x)^{n+1} (d x (b c-a d) (a d+b c (n+3))+c (b c (n+2) (a d+b c (n+3))-a d (a d+b c (3 n+5))))}{b^2 d^3 (n+1) (n+2) (c+d x) (b c-a d)}-\frac {c^2 (a+b x)^{n+1} (3 a d-b c (n+3)) \, _2F_1\left (1,n+1;n+2;-\frac {d (a+b x)}{b c-a d}\right )}{d^3 (n+1) (b c-a d)^2}+\frac {x^2 (a+b x)^{n+1}}{b d (n+2) (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 102
Rule 151
Rubi steps
\begin {align*} \int \frac {x^3 (a+b x)^n}{(c+d x)^2} \, dx &=\frac {x^2 (a+b x)^{1+n}}{b d (2+n) (c+d x)}+\frac {\int \frac {x (a+b x)^n (-2 a c+(-a d-b c (3+n)) x)}{(c+d x)^2} \, dx}{b d (2+n)}\\ &=\frac {x^2 (a+b x)^{1+n}}{b d (2+n) (c+d x)}-\frac {(a+b x)^{1+n} (c (b c (2+n) (a d+b c (3+n))-a d (a d+b c (5+3 n)))+d (b c-a d) (a d+b c (3+n)) x)}{b^2 d^3 (b c-a d) (1+n) (2+n) (c+d x)}-\frac {\left (c^2 (3 a d-b c (3+n))\right ) \int \frac {(a+b x)^n}{c+d x} \, dx}{d^3 (b c-a d)}\\ &=\frac {x^2 (a+b x)^{1+n}}{b d (2+n) (c+d x)}-\frac {(a+b x)^{1+n} (c (b c (2+n) (a d+b c (3+n))-a d (a d+b c (5+3 n)))+d (b c-a d) (a d+b c (3+n)) x)}{b^2 d^3 (b c-a d) (1+n) (2+n) (c+d x)}-\frac {c^2 (3 a d-b c (3+n)) (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;-\frac {d (a+b x)}{b c-a d}\right )}{d^3 (b c-a d)^2 (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 176, normalized size = 0.87 \begin {gather*} \frac {(a+b x)^{1+n} \left (\frac {x^2}{c+d x}+\frac {a^2 d^2 (c+d x)-b^2 c^2 (3+n) (c (2+n)+d x)+a b c d (c (3+2 n)+d (2+n) x)}{b d^2 (b c-a d) (1+n) (c+d x)}+\frac {b c^2 (2+n) (-3 a d+b c (3+n)) \, _2F_1\left (1,1+n;2+n;\frac {d (a+b x)}{-b c+a d}\right )}{d^2 (b c-a d)^2 (1+n)}\right )}{b d (2+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \left (b x +a \right )^{n}}{\left (d x +c \right )^{2}}\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \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 = -0.00 \begin {gather*} \int \frac {x^3\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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