Optimal. Leaf size=62 \[ -\frac {c (a+b x)^{1+n}}{a x}-\frac {(a d+b c n) (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {b x}{a}\right )}{a^2 (1+n)} \]
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Rubi [A]
time = 0.01, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 67}
\begin {gather*} -\frac {(a+b x)^{n+1} (a d+b c n) \, _2F_1\left (1,n+1;n+2;\frac {b x}{a}+1\right )}{a^2 (n+1)}-\frac {c (a+b x)^{n+1}}{a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 79
Rubi steps
\begin {align*} \int \frac {(a+b x)^n (c+d x)}{x^2} \, dx &=-\frac {c (a+b x)^{1+n}}{a x}+\frac {(a d+b c n) \int \frac {(a+b x)^n}{x} \, dx}{a}\\ &=-\frac {c (a+b x)^{1+n}}{a x}-\frac {(a d+b c n) (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {b x}{a}\right )}{a^2 (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 55, normalized size = 0.89 \begin {gather*} -\frac {(a+b x)^{1+n} \left (a c (1+n)+(a d+b c n) x \, _2F_1\left (1,1+n;2+n;1+\frac {b x}{a}\right )\right )}{a^2 (1+n) x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{n} \left (d x +c \right )}{x^{2}}\, dx\]
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. 493 vs.
\(2 (49) = 98\).
time = 3.34, size = 493, normalized size = 7.95 \begin {gather*} \frac {b^{n} c n^{2} \left (\frac {a}{b} + x\right )^{n} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{x \Gamma \left (n + 2\right )} + \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 )}{x \Gamma \left (n + 2\right )} - \frac {b^{n} c n \left (\frac {a}{b} + x\right )^{n} \Gamma \left (n + 1\right )}{x \Gamma \left (n + 2\right )} - \frac {b^{n} c \left (\frac {a}{b} + x\right )^{n} \Gamma \left (n + 1\right )}{x \Gamma \left (n + 2\right )} - \frac {b^{n} d 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} d \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 b^{n} c n^{2} \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 n \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 n \left (\frac {a}{b} + x\right )^{n} \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} - \frac {b b^{n} c \left (\frac {a}{b} + x\right )^{n} \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} - \frac {b b^{n} d 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} d 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^{2} b^{n} c n^{2} \left (\frac {a}{b} + x\right )^{2} \left (\frac {a}{b} + x\right )^{n} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a^{2} x \Gamma \left (n + 2\right )} - \frac {b^{2} b^{n} c n \left (\frac {a}{b} + x\right )^{2} \left (\frac {a}{b} + x\right )^{n} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a^{2} x \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 = -0.02 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^n\,\left (c+d\,x\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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