Optimal. Leaf size=131 \[ \frac {d \left (a^2 d^2-3 a b c d+3 b^2 c^2\right ) (a+b x)^{n+1}}{b^3 (n+1)}+\frac {d^2 (3 b c-2 a d) (a+b x)^{n+2}}{b^3 (n+2)}+\frac {d^3 (a+b x)^{n+3}}{b^3 (n+3)}-\frac {c^3 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {b x}{a}+1\right )}{a (n+1)} \]
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Rubi [A] time = 0.06, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {88, 65} \[ \frac {d \left (a^2 d^2-3 a b c d+3 b^2 c^2\right ) (a+b x)^{n+1}}{b^3 (n+1)}+\frac {d^2 (3 b c-2 a d) (a+b x)^{n+2}}{b^3 (n+2)}+\frac {d^3 (a+b x)^{n+3}}{b^3 (n+3)}-\frac {c^3 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {b x}{a}+1\right )}{a (n+1)} \]
Antiderivative was successfully verified.
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Rule 65
Rule 88
Rubi steps
\begin {align*} \int \frac {(a+b x)^n (c+d x)^3}{x} \, dx &=\int \left (\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) (a+b x)^n}{b^2}+\frac {c^3 (a+b x)^n}{x}+\frac {d^2 (3 b c-2 a d) (a+b x)^{1+n}}{b^2}+\frac {d^3 (a+b x)^{2+n}}{b^2}\right ) \, dx\\ &=\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) (a+b x)^{1+n}}{b^3 (1+n)}+\frac {d^2 (3 b c-2 a d) (a+b x)^{2+n}}{b^3 (2+n)}+\frac {d^3 (a+b x)^{3+n}}{b^3 (3+n)}+c^3 \int \frac {(a+b x)^n}{x} \, dx\\ &=\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) (a+b x)^{1+n}}{b^3 (1+n)}+\frac {d^2 (3 b c-2 a d) (a+b x)^{2+n}}{b^3 (2+n)}+\frac {d^3 (a+b x)^{3+n}}{b^3 (3+n)}-\frac {c^3 (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {b x}{a}\right )}{a (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 117, normalized size = 0.89 \[ (a+b x)^{n+1} \left (\frac {d \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3 (n+1)}+\frac {d^2 (a+b x) (3 b c-2 a d)}{b^3 (n+2)}+\frac {d^3 (a+b x)^2}{b^3 (n+3)}-\frac {c^3 \, _2F_1\left (1,n+1;n+2;\frac {a+b x}{a}\right )}{a n+a}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} {\left (b x + a\right )}^{n}}{x}, x\right ) \]
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 \[ \int \frac {{\left (d x + c\right )}^{3} {\left (b x + a\right )}^{n}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right )^{3} \left (b x +a \right )^{n}}{x}\, dx \]
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 \[ \frac {3 \, {\left (b x + a\right )}^{n + 1} c^{2} d}{b {\left (n + 1\right )}} + \int \frac {{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + c^{3}\right )} {\left (b x + a\right )}^{n}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,x\right )}^n\,{\left (c+d\,x\right )}^3}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 10.03, size = 993, normalized size = 7.58 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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