Optimal. Leaf size=99 \[ \frac {c d x^{1+m}}{b (1+m)}+\frac {d (b c-a d) x^{1+m}}{b^2 (1+m)}+\frac {d^2 x^{2+m}}{b (2+m)}+\frac {(b c-a d)^2 x^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )}{a b^2 (1+m)} \]
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Rubi [A]
time = 0.04, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {90, 66, 45}
\begin {gather*} \frac {x^{m+1} (b c-a d)^2 \, _2F_1\left (1,m+1;m+2;-\frac {b x}{a}\right )}{a b^2 (m+1)}+\frac {d x^{m+1} (b c-a d)}{b^2 (m+1)}+\frac {c d x^{m+1}}{b (m+1)}+\frac {d^2 x^{m+2}}{b (m+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 66
Rule 90
Rubi steps
\begin {align*} \int \frac {x^m (c+d x)^2}{a+b x} \, dx &=\int \left (\frac {d (b c-a d) x^m}{b^2}+\frac {(b c-a d)^2 x^m}{b^2 (a+b x)}+\frac {d x^m (c+d x)}{b}\right ) \, dx\\ &=\frac {d (b c-a d) x^{1+m}}{b^2 (1+m)}+\frac {d \int x^m (c+d x) \, dx}{b}+\frac {(b c-a d)^2 \int \frac {x^m}{a+b x} \, dx}{b^2}\\ &=\frac {d (b c-a d) x^{1+m}}{b^2 (1+m)}+\frac {(b c-a d)^2 x^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )}{a b^2 (1+m)}+\frac {d \int \left (c x^m+d x^{1+m}\right ) \, dx}{b}\\ &=\frac {c d x^{1+m}}{b (1+m)}+\frac {d (b c-a d) x^{1+m}}{b^2 (1+m)}+\frac {d^2 x^{2+m}}{b (2+m)}+\frac {(b c-a d)^2 x^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )}{a b^2 (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 77, normalized size = 0.78 \begin {gather*} \frac {x^{1+m} \left (a d (2 b c (2+m)-a d (2+m)+b d (1+m) x)+(b c-a d)^2 (2+m) \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )\right )}{a b^2 (1+m) (2+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{m} \left (d x +c \right )^{2}}{b x +a}\, dx\]
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 [C] Result contains complex when optimal does not.
time = 1.83, size = 219, normalized size = 2.21 \begin {gather*} \frac {c^{2} m x x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )} + \frac {c^{2} x x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )} + \frac {2 c d m x^{2} x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a \Gamma \left (m + 3\right )} + \frac {4 c d x^{2} x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a \Gamma \left (m + 3\right )} + \frac {d^{2} m x^{3} x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 3\right ) \Gamma \left (m + 3\right )}{a \Gamma \left (m + 4\right )} + \frac {3 d^{2} x^{3} x^{m} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 3\right ) \Gamma \left (m + 3\right )}{a \Gamma \left (m + 4\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 {x^m\,{\left (c+d\,x\right )}^2}{a+b\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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