Optimal. Leaf size=358 \[ -\frac {5 a d^2 \left (3 b^3 c-14 a^3 d\right ) (a+b x)^{n+5}}{b^9 (n+5)}+\frac {d^2 \left (3 b^3 c-56 a^3 d\right ) (a+b x)^{n+6}}{b^9 (n+6)}+\frac {28 a^2 d^3 (a+b x)^{n+7}}{b^9 (n+7)}-\frac {a d \left (8 a^6 d^2-15 a^3 b^3 c d+6 b^6 c^2\right ) (a+b x)^{n+2}}{b^9 (n+2)}+\frac {d \left (28 a^6 d^2-30 a^3 b^3 c d+3 b^6 c^2\right ) (a+b x)^{n+3}}{b^9 (n+3)}+\frac {2 a^2 d^2 \left (15 b^3 c-28 a^3 d\right ) (a+b x)^{n+4}}{b^9 (n+4)}+\frac {a^2 d \left (a^6 d^2-3 a^3 b^3 c d+3 b^6 c^2\right ) (a+b x)^{n+1}}{b^9 (n+1)}-\frac {8 a d^3 (a+b x)^{n+8}}{b^9 (n+8)}+\frac {d^3 (a+b x)^{n+9}}{b^9 (n+9)}-\frac {c^3 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {b x}{a}+1\right )}{a (n+1)} \]
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Rubi [A] time = 0.22, antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1620, 65} \[ \frac {a^2 d \left (-3 a^3 b^3 c d+a^6 d^2+3 b^6 c^2\right ) (a+b x)^{n+1}}{b^9 (n+1)}-\frac {a d \left (-15 a^3 b^3 c d+8 a^6 d^2+6 b^6 c^2\right ) (a+b x)^{n+2}}{b^9 (n+2)}+\frac {d \left (-30 a^3 b^3 c d+28 a^6 d^2+3 b^6 c^2\right ) (a+b x)^{n+3}}{b^9 (n+3)}+\frac {2 a^2 d^2 \left (15 b^3 c-28 a^3 d\right ) (a+b x)^{n+4}}{b^9 (n+4)}-\frac {5 a d^2 \left (3 b^3 c-14 a^3 d\right ) (a+b x)^{n+5}}{b^9 (n+5)}+\frac {d^2 \left (3 b^3 c-56 a^3 d\right ) (a+b x)^{n+6}}{b^9 (n+6)}+\frac {28 a^2 d^3 (a+b x)^{n+7}}{b^9 (n+7)}-\frac {8 a d^3 (a+b x)^{n+8}}{b^9 (n+8)}+\frac {d^3 (a+b x)^{n+9}}{b^9 (n+9)}-\frac {c^3 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {b x}{a}+1\right )}{a (n+1)} \]
Antiderivative was successfully verified.
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Rule 65
Rule 1620
Rubi steps
\begin {align*} \int \frac {(a+b x)^n \left (c+d x^3\right )^3}{x} \, dx &=\int \left (\frac {a^2 d \left (3 b^6 c^2-3 a^3 b^3 c d+a^6 d^2\right ) (a+b x)^n}{b^8}+\frac {c^3 (a+b x)^n}{x}-\frac {a d \left (6 b^6 c^2-15 a^3 b^3 c d+8 a^6 d^2\right ) (a+b x)^{1+n}}{b^8}+\frac {d \left (3 b^6 c^2-30 a^3 b^3 c d+28 a^6 d^2\right ) (a+b x)^{2+n}}{b^8}-\frac {2 a^2 d^2 \left (-15 b^3 c+28 a^3 d\right ) (a+b x)^{3+n}}{b^8}+\frac {5 a d^2 \left (-3 b^3 c+14 a^3 d\right ) (a+b x)^{4+n}}{b^8}+\frac {d^2 \left (3 b^3 c-56 a^3 d\right ) (a+b x)^{5+n}}{b^8}+\frac {28 a^2 d^3 (a+b x)^{6+n}}{b^8}-\frac {8 a d^3 (a+b x)^{7+n}}{b^8}+\frac {d^3 (a+b x)^{8+n}}{b^8}\right ) \, dx\\ &=\frac {a^2 d \left (3 b^6 c^2-3 a^3 b^3 c d+a^6 d^2\right ) (a+b x)^{1+n}}{b^9 (1+n)}-\frac {a d \left (6 b^6 c^2-15 a^3 b^3 c d+8 a^6 d^2\right ) (a+b x)^{2+n}}{b^9 (2+n)}+\frac {d \left (3 b^6 c^2-30 a^3 b^3 c d+28 a^6 d^2\right ) (a+b x)^{3+n}}{b^9 (3+n)}+\frac {2 a^2 d^2 \left (15 b^3 c-28 a^3 d\right ) (a+b x)^{4+n}}{b^9 (4+n)}-\frac {5 a d^2 \left (3 b^3 c-14 a^3 d\right ) (a+b x)^{5+n}}{b^9 (5+n)}+\frac {d^2 \left (3 b^3 c-56 a^3 d\right ) (a+b x)^{6+n}}{b^9 (6+n)}+\frac {28 a^2 d^3 (a+b x)^{7+n}}{b^9 (7+n)}-\frac {8 a d^3 (a+b x)^{8+n}}{b^9 (8+n)}+\frac {d^3 (a+b x)^{9+n}}{b^9 (9+n)}+c^3 \int \frac {(a+b x)^n}{x} \, dx\\ &=\frac {a^2 d \left (3 b^6 c^2-3 a^3 b^3 c d+a^6 d^2\right ) (a+b x)^{1+n}}{b^9 (1+n)}-\frac {a d \left (6 b^6 c^2-15 a^3 b^3 c d+8 a^6 d^2\right ) (a+b x)^{2+n}}{b^9 (2+n)}+\frac {d \left (3 b^6 c^2-30 a^3 b^3 c d+28 a^6 d^2\right ) (a+b x)^{3+n}}{b^9 (3+n)}+\frac {2 a^2 d^2 \left (15 b^3 c-28 a^3 d\right ) (a+b x)^{4+n}}{b^9 (4+n)}-\frac {5 a d^2 \left (3 b^3 c-14 a^3 d\right ) (a+b x)^{5+n}}{b^9 (5+n)}+\frac {d^2 \left (3 b^3 c-56 a^3 d\right ) (a+b x)^{6+n}}{b^9 (6+n)}+\frac {28 a^2 d^3 (a+b x)^{7+n}}{b^9 (7+n)}-\frac {8 a d^3 (a+b x)^{8+n}}{b^9 (8+n)}+\frac {d^3 (a+b x)^{9+n}}{b^9 (9+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 = 0.35, size = 332, normalized size = 0.93 \[ (a+b x)^{n+1} \left (\frac {d^2 (a+b x)^5 \left (3 b^3 c-56 a^3 d\right )}{b^9 (n+6)}+\frac {5 a d^2 (a+b x)^4 \left (14 a^3 d-3 b^3 c\right )}{b^9 (n+5)}+\frac {28 a^2 d^3 (a+b x)^6}{b^9 (n+7)}+\frac {d (a+b x)^2 \left (28 a^6 d^2-30 a^3 b^3 c d+3 b^6 c^2\right )}{b^9 (n+3)}-\frac {a d (a+b x) \left (8 a^6 d^2-15 a^3 b^3 c d+6 b^6 c^2\right )}{b^9 (n+2)}+\frac {2 a^2 d^2 (a+b x)^3 \left (15 b^3 c-28 a^3 d\right )}{b^9 (n+4)}+\frac {a^2 d \left (a^6 d^2-3 a^3 b^3 c d+3 b^6 c^2\right )}{b^9 (n+1)}+\frac {d^3 (a+b x)^8}{b^9 (n+9)}-\frac {8 a d^3 (a+b x)^7}{b^9 (n+8)}-\frac {c^3 \, _2F_1\left (1,n+1;n+2;\frac {a+b x}{a}\right )}{a n+a}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d^{3} x^{9} + 3 \, c d^{2} x^{6} + 3 \, c^{2} d x^{3} + c^{3}\right )} {\left (b x + a\right )}^{n}}{x}, x\right ) \]
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 \[ \int \frac {{\left (d x^{3} + c\right )}^{3} {\left (b x + a\right )}^{n}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \,x^{3}+c \right )^{3} \left (b x +a \right )^{n}}{x}\, dx \]
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 \[ \int \frac {{\left (d x^{3} + c\right )}^{3} {\left (b x + a\right )}^{n}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (d\,x^3+c\right )}^3\,{\left (a+b\,x\right )}^n}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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