Optimal. Leaf size=329 \[ -\frac{(c+d x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right ) \left (a^2 b d (n+2) (6 c D+C d (n+1))+a^3 \left (-d^2\right ) D \left (n^2+5 n+6\right )-a b^2 \left (B d^2 n (n+1)+6 c^2 D+4 c C d (n+1)\right )+b^3 \left (-A d^2 (1-n) n+2 B c d n+2 c^2 C\right )\right )}{2 b^3 (n+1) (b c-a d)^3}-\frac{(c+d x)^{n+1} \left (a^2 b (6 c D+C d (n+3))+a^3 (-d) D (n+5)-a b^2 (B d (n+1)+4 c C)+b^3 (2 B c-A d (1-n))\right )}{2 b^3 (a+b x) (b c-a d)^2}-\frac{(c+d x)^{n+1} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{2 b^3 (a+b x)^2 (b c-a d)}+\frac{D (c+d x)^{n+1}}{b^3 d (n+1)} \]
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Rubi [A] time = 0.61896, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1621, 949, 80, 68} \[ -\frac{(c+d x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right ) \left (a^2 b d (n+2) (6 c D+C d (n+1))+a^3 \left (-d^2\right ) D \left (n^2+5 n+6\right )-a b^2 \left (B d^2 n (n+1)+6 c^2 D+4 c C d (n+1)\right )+b^3 \left (-A d^2 (1-n) n+2 B c d n+2 c^2 C\right )\right )}{2 b^3 (n+1) (b c-a d)^3}-\frac{(c+d x)^{n+1} \left (a^2 b (6 c D+C d (n+3))+a^3 (-d) D (n+5)-a b^2 (B d (n+1)+4 c C)+b^3 (2 B c-A d (1-n))\right )}{2 b^3 (a+b x) (b c-a d)^2}-\frac{(c+d x)^{n+1} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{2 b^3 (a+b x)^2 (b c-a d)}+\frac{D (c+d x)^{n+1}}{b^3 d (n+1)} \]
Antiderivative was successfully verified.
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Rule 1621
Rule 949
Rule 80
Rule 68
Rubi steps
\begin{align*} \int \frac{(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^3} \, dx &=-\frac{\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) (c+d x)^{1+n}}{2 b^3 (b c-a d) (a+b x)^2}-\frac{\int \frac{(c+d x)^n \left (-2 B c+A d (1-n)+\frac{a^3 d D (1+n)}{b^3}+\frac{a (2 c C+B d (1+n))}{b}-\frac{a^2 (2 c D+C d (1+n))}{b^2}-\frac{2 (b c-a d) (b C-a D) x}{b^2}-2 \left (c-\frac{a d}{b}\right ) D x^2\right )}{(a+b x)^2} \, dx}{2 (b c-a d)}\\ &=-\frac{\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) (c+d x)^{1+n}}{2 b^3 (b c-a d) (a+b x)^2}-\frac{\left (b^3 (2 B c-A d (1-n))-a^3 d D (5+n)-a b^2 (4 c C+B d (1+n))+a^2 b (6 c D+C d (3+n))\right ) (c+d x)^{1+n}}{2 b^3 (b c-a d)^2 (a+b x)}+\frac{\int \frac{(c+d x)^n \left (2 c^2 C+2 B c d n-A d^2 (1-n) n-\frac{a^3 d^2 D \left (4+5 n+n^2\right )}{b^3}-\frac{a \left (4 c^2 D+4 c C d (1+n)+B d^2 n (1+n)\right )}{b}+\frac{a^2 d \left (2 c D (4+3 n)+C d \left (2+3 n+n^2\right )\right )}{b^2}+\frac{2 (b c-a d)^2 D x}{b^2}\right )}{a+b x} \, dx}{2 (b c-a d)^2}\\ &=\frac{D (c+d x)^{1+n}}{b^3 d (1+n)}-\frac{\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) (c+d x)^{1+n}}{2 b^3 (b c-a d) (a+b x)^2}-\frac{\left (b^3 (2 B c-A d (1-n))-a^3 d D (5+n)-a b^2 (4 c C+B d (1+n))+a^2 b (6 c D+C d (3+n))\right ) (c+d x)^{1+n}}{2 b^3 (b c-a d)^2 (a+b x)}+\frac{\left (b^3 \left (2 c^2 C+2 B c d n-A d^2 (1-n) n\right )-a^3 d^2 D \left (6+5 n+n^2\right )+a^2 b d (2+n) (6 c D+C d (1+n))-a b^2 \left (6 c^2 D+4 c C d (1+n)+B d^2 n (1+n)\right )\right ) \int \frac{(c+d x)^n}{a+b x} \, dx}{2 b^3 (b c-a d)^2}\\ &=\frac{D (c+d x)^{1+n}}{b^3 d (1+n)}-\frac{\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) (c+d x)^{1+n}}{2 b^3 (b c-a d) (a+b x)^2}-\frac{\left (b^3 (2 B c-A d (1-n))-a^3 d D (5+n)-a b^2 (4 c C+B d (1+n))+a^2 b (6 c D+C d (3+n))\right ) (c+d x)^{1+n}}{2 b^3 (b c-a d)^2 (a+b x)}-\frac{\left (b^3 \left (2 c^2 C+2 B c d n-A d^2 (1-n) n\right )-a^3 d^2 D \left (6+5 n+n^2\right )+a^2 b d (2+n) (6 c D+C d (1+n))-a b^2 \left (6 c^2 D+4 c C d (1+n)+B d^2 n (1+n)\right )\right ) (c+d x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{b (c+d x)}{b c-a d}\right )}{2 b^3 (b c-a d)^3 (1+n)}\\ \end{align*}
Mathematica [A] time = 0.199022, size = 188, normalized size = 0.57 \[ \frac{(c+d x)^{n+1} \left (-\frac{d^2 \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \, _2F_1\left (3,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{(b c-a d)^3}+\frac{d \left (3 a^2 D-2 a b C+b^2 B\right ) \, _2F_1\left (2,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{(b c-a d)^2}-\frac{(b C-3 a D) \, _2F_1\left (1,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{b c-a d}+\frac{D}{d}\right )}{b^3 (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.06, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx+c \right ) ^{n} \left ( D{x}^{3}+C{x}^{2}+Bx+A \right ) }{ \left ( bx+a \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (D x^{3} + C x^{2} + B x + A\right )}{\left (d x + c\right )}^{n}}{{\left (b x + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c + d x\right )^{n} \left (A + B x + C x^{2} + D x^{3}\right )}{\left (a + b x\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (D x^{3} + C x^{2} + B x + A\right )}{\left (d x + c\right )}^{n}}{{\left (b x + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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