Optimal. Leaf size=388 \[ \frac{(e f-d g) (d+e x)^{m+1} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F_1\left (m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e^2 (m+1) \sqrt{a+b x+c x^2}}+\frac{g (d+e x)^{m+2} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F_1\left (m+2;\frac{1}{2},\frac{1}{2};m+3;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e^2 (m+2) \sqrt{a+b x+c x^2}} \]
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Rubi [A] time = 0.346535, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {843, 759, 133} \[ \frac{(e f-d g) (d+e x)^{m+1} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F_1\left (m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e^2 (m+1) \sqrt{a+b x+c x^2}}+\frac{g (d+e x)^{m+2} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F_1\left (m+2;\frac{1}{2},\frac{1}{2};m+3;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e^2 (m+2) \sqrt{a+b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 843
Rule 759
Rule 133
Rubi steps
\begin{align*} \int \frac{(d+e x)^m (f+g x)}{\sqrt{a+b x+c x^2}} \, dx &=\frac{g \int \frac{(d+e x)^{1+m}}{\sqrt{a+b x+c x^2}} \, dx}{e}+\frac{(e f-d g) \int \frac{(d+e x)^m}{\sqrt{a+b x+c x^2}} \, dx}{e}\\ &=\frac{\left (g \sqrt{1-\frac{d+e x}{d-\frac{\left (b-\sqrt{b^2-4 a c}\right ) e}{2 c}}} \sqrt{1-\frac{d+e x}{d-\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c}}}\right ) \operatorname{Subst}\left (\int \frac{x^{1+m}}{\sqrt{1-\frac{2 c x}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}} \sqrt{1-\frac{2 c x}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}}} \, dx,x,d+e x\right )}{e^2 \sqrt{a+b x+c x^2}}+\frac{\left ((e f-d g) \sqrt{1-\frac{d+e x}{d-\frac{\left (b-\sqrt{b^2-4 a c}\right ) e}{2 c}}} \sqrt{1-\frac{d+e x}{d-\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c}}}\right ) \operatorname{Subst}\left (\int \frac{x^m}{\sqrt{1-\frac{2 c x}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}} \sqrt{1-\frac{2 c x}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}}} \, dx,x,d+e x\right )}{e^2 \sqrt{a+b x+c x^2}}\\ &=\frac{(e f-d g) (d+e x)^{1+m} \sqrt{1-\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} F_1\left (1+m;\frac{1}{2},\frac{1}{2};2+m;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e^2 (1+m) \sqrt{a+b x+c x^2}}+\frac{g (d+e x)^{2+m} \sqrt{1-\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} F_1\left (2+m;\frac{1}{2},\frac{1}{2};3+m;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e^2 (2+m) \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [F] time = 0.742372, size = 0, normalized size = 0. \[ \int \frac{(d+e x)^m (f+g x)}{\sqrt{a+b x+c x^2}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 1.243, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ex+d \right ) ^{m} \left ( gx+f \right ){\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}\, 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 (g x + f\right )}{\left (e x + d\right )}^{m}}{\sqrt{c x^{2} + b x + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (g x + f\right )}{\left (e x + d\right )}^{m}}{\sqrt{c x^{2} + b x + a}}, x\right ) \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 (d + e x\right )^{m} \left (f + g x\right )}{\sqrt{a + b x + c x^{2}}}\, 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 (g x + f\right )}{\left (e x + d\right )}^{m}}{\sqrt{c x^{2} + b x + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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