Optimal. Leaf size=99 \[ \frac {a^2 d (a+b x)^{1+n}}{b^3 (1+n)}-\frac {2 a d (a+b x)^{2+n}}{b^3 (2+n)}+\frac {d (a+b x)^{3+n}}{b^3 (3+n)}-\frac {c (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {b x}{a}\right )}{a (1+n)} \]
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Rubi [A]
time = 0.04, antiderivative size = 99, 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 = {1634, 67}
\begin {gather*} \frac {a^2 d (a+b x)^{n+1}}{b^3 (n+1)}-\frac {2 a d (a+b x)^{n+2}}{b^3 (n+2)}+\frac {d (a+b x)^{n+3}}{b^3 (n+3)}-\frac {c (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {b x}{a}+1\right )}{a (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 1634
Rubi steps
\begin {align*} \int \frac {(a+b x)^n \left (c+d x^3\right )}{x} \, dx &=\int \left (\frac {a^2 d (a+b x)^n}{b^2}+\frac {c (a+b x)^n}{x}-\frac {2 a d (a+b x)^{1+n}}{b^2}+\frac {d (a+b x)^{2+n}}{b^2}\right ) \, dx\\ &=\frac {a^2 d (a+b x)^{1+n}}{b^3 (1+n)}-\frac {2 a d (a+b x)^{2+n}}{b^3 (2+n)}+\frac {d (a+b x)^{3+n}}{b^3 (3+n)}+c \int \frac {(a+b x)^n}{x} \, dx\\ &=\frac {a^2 d (a+b x)^{1+n}}{b^3 (1+n)}-\frac {2 a d (a+b x)^{2+n}}{b^3 (2+n)}+\frac {d (a+b x)^{3+n}}{b^3 (3+n)}-\frac {c (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 = 0.08, size = 94, normalized size = 0.95 \begin {gather*} \frac {(a+b x)^{1+n} \left (a d \left (2 a^2-2 a b (1+n) x+b^2 \left (2+3 n+n^2\right ) x^2\right )-b^3 c \left (6+5 n+n^2\right ) \, _2F_1\left (1,1+n;2+n;1+\frac {b x}{a}\right )\right )}{a b^3 (1+n) (2+n) (3+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{n} \left (d \,x^{3}+c \right )}{x}\, dx\]
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 \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 598 vs.
\(2 (83) = 166\).
time = 2.99, size = 741, normalized size = 7.48 \begin {gather*} - \frac {b^{n} c n \left (\frac {a}{b} + x\right )^{n} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{\Gamma \left (n + 2\right )} - \frac {b^{n} c \left (\frac {a}{b} + x\right )^{n} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{\Gamma \left (n + 2\right )} + d \left (\begin {cases} \frac {a^{n} x^{3}}{3} & \text {for}\: b = 0 \\\frac {2 a^{2} \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {3 a^{2}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {4 a b x \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {4 a b x}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {2 b^{2} x^{2} \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} & \text {for}\: n = -3 \\- \frac {2 a^{2} \log {\left (\frac {a}{b} + x \right )}}{a b^{3} + b^{4} x} - \frac {2 a^{2}}{a b^{3} + b^{4} x} - \frac {2 a b x \log {\left (\frac {a}{b} + x \right )}}{a b^{3} + b^{4} x} + \frac {b^{2} x^{2}}{a b^{3} + b^{4} x} & \text {for}\: n = -2 \\\frac {a^{2} \log {\left (\frac {a}{b} + x \right )}}{b^{3}} - \frac {a x}{b^{2}} + \frac {x^{2}}{2 b} & \text {for}\: n = -1 \\\frac {2 a^{3} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} - \frac {2 a^{2} b n x \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {a b^{2} n^{2} x^{2} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {a b^{2} n x^{2} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {b^{3} n^{2} x^{3} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {3 b^{3} n x^{3} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac {2 b^{3} x^{3} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} & \text {otherwise} \end {cases}\right ) - \frac {b b^{n} c n x \left (\frac {a}{b} + x\right )^{n} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} - \frac {b b^{n} c x \left (\frac {a}{b} + x\right )^{n} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} \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 = -0.01 \begin {gather*} \int \frac {\left (d\,x^3+c\right )\,{\left (a+b\,x\right )}^n}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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