Optimal. Leaf size=118 \[ -\frac {d^2 \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b x^2}{a}+1\right )}{2 a (p+1)}+2 d e x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )+\frac {e^2 \left (a+b x^2\right )^{p+1}}{2 b (p+1)} \]
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Rubi [A] time = 0.09, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1652, 446, 80, 65, 12, 246, 245} \[ -\frac {d^2 \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b x^2}{a}+1\right )}{2 a (p+1)}+2 d e x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )+\frac {e^2 \left (a+b x^2\right )^{p+1}}{2 b (p+1)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 80
Rule 245
Rule 246
Rule 446
Rule 1652
Rubi steps
\begin {align*} \int \frac {(d+e x)^2 \left (a+b x^2\right )^p}{x} \, dx &=\int 2 d e \left (a+b x^2\right )^p \, dx+\int \frac {\left (a+b x^2\right )^p \left (d^2+e^2 x^2\right )}{x} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^p \left (d^2+e^2 x\right )}{x} \, dx,x,x^2\right )+(2 d e) \int \left (a+b x^2\right )^p \, dx\\ &=\frac {e^2 \left (a+b x^2\right )^{1+p}}{2 b (1+p)}+\frac {1}{2} d^2 \operatorname {Subst}\left (\int \frac {(a+b x)^p}{x} \, dx,x,x^2\right )+\left (2 d e \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int \left (1+\frac {b x^2}{a}\right )^p \, dx\\ &=\frac {e^2 \left (a+b x^2\right )^{1+p}}{2 b (1+p)}+2 d e x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )-\frac {d^2 \left (a+b x^2\right )^{1+p} \, _2F_1\left (1,1+p;2+p;1+\frac {b x^2}{a}\right )}{2 a (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 101, normalized size = 0.86 \[ \frac {1}{2} \left (a+b x^2\right )^p \left (\frac {\left (a+b x^2\right ) \left (a e^2-b d^2 \, _2F_1\left (1,p+1;p+2;\frac {b x^2}{a}+1\right )\right )}{a b (p+1)}+4 d e x \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} {\left (b x^{2} + a\right )}^{p}}{x}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{2} {\left (b x^{2} + a\right )}^{p}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x +d \right )^{2} \left (b \,x^{2}+a \right )^{p}}{x}\, dx \]
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 \[ \int \frac {{\left (e x + d\right )}^{2} {\left (b x^{2} + a\right )}^{p}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,x^2+a\right )}^p\,{\left (d+e\,x\right )}^2}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.06, size = 109, normalized size = 0.92 \[ 2 a^{p} d e x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - p \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )} - \frac {b^{p} d^{2} x^{2 p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{2 \Gamma \left (1 - p\right )} + e^{2} \left (\begin {cases} \frac {a^{p} x^{2}}{2} & \text {for}\: b = 0 \\\frac {\begin {cases} \frac {\left (a + b x^{2}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (a + b x^{2} \right )} & \text {otherwise} \end {cases}}{2 b} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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