Optimal. Leaf size=281 \[ \frac{A (e x)^{m+1} \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )^{3/2} \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )^{3/2} F_1\left (m+1;\frac{3}{2},\frac{3}{2};m+2;-\frac{2 c x}{b-\sqrt{b^2-4 a c}},-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{e (m+1) \left (a+b x+c x^2\right )^{3/2}}+\frac{B (e x)^{m+2} \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )^{3/2} \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )^{3/2} F_1\left (m+2;\frac{3}{2},\frac{3}{2};m+3;-\frac{2 c x}{b-\sqrt{b^2-4 a c}},-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{e^2 (m+2) \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.406844, antiderivative size = 281, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {843, 759, 133} \[ \frac{A (e x)^{m+1} \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )^{3/2} \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )^{3/2} F_1\left (m+1;\frac{3}{2},\frac{3}{2};m+2;-\frac{2 c x}{b-\sqrt{b^2-4 a c}},-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{e (m+1) \left (a+b x+c x^2\right )^{3/2}}+\frac{B (e x)^{m+2} \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )^{3/2} \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )^{3/2} F_1\left (m+2;\frac{3}{2},\frac{3}{2};m+3;-\frac{2 c x}{b-\sqrt{b^2-4 a c}},-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{e^2 (m+2) \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 843
Rule 759
Rule 133
Rubi steps
\begin{align*} \int \frac{(e x)^m (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=A \int \frac{(e x)^m}{\left (a+b x+c x^2\right )^{3/2}} \, dx+\frac{B \int \frac{(e x)^{1+m}}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{e}\\ &=\frac{\left (B \left (1+\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )^{3/2} \left (1+\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{x^{1+m}}{\left (1+\frac{2 c x}{\left (b-\sqrt{b^2-4 a c}\right ) e}\right )^{3/2} \left (1+\frac{2 c x}{\left (b+\sqrt{b^2-4 a c}\right ) e}\right )^{3/2}} \, dx,x,e x\right )}{e^2 \left (a+b x+c x^2\right )^{3/2}}+\frac{\left (A \left (1+\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )^{3/2} \left (1+\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{x^m}{\left (1+\frac{2 c x}{\left (b-\sqrt{b^2-4 a c}\right ) e}\right )^{3/2} \left (1+\frac{2 c x}{\left (b+\sqrt{b^2-4 a c}\right ) e}\right )^{3/2}} \, dx,x,e x\right )}{e \left (a+b x+c x^2\right )^{3/2}}\\ &=\frac{A (e x)^{1+m} \left (1+\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )^{3/2} \left (1+\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )^{3/2} F_1\left (1+m;\frac{3}{2},\frac{3}{2};2+m;-\frac{2 c x}{b-\sqrt{b^2-4 a c}},-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{e (1+m) \left (a+b x+c x^2\right )^{3/2}}+\frac{B (e x)^{2+m} \left (1+\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )^{3/2} \left (1+\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )^{3/2} F_1\left (2+m;\frac{3}{2},\frac{3}{2};3+m;-\frac{2 c x}{b-\sqrt{b^2-4 a c}},-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{e^2 (2+m) \left (a+b x+c x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.344043, size = 272, normalized size = 0.97 \[ \frac{x (e x)^m \left (\sqrt{b^2-4 a c}-b-2 c x\right ) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x}{b-\sqrt{b^2-4 a c}}} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}+b}\right )^{3/2} \left (A (m+2) F_1\left (m+1;\frac{3}{2},\frac{3}{2};m+2;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )+B (m+1) x F_1\left (m+2;\frac{3}{2},\frac{3}{2};m+3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )\right )}{(m+1) (m+2) \left (\sqrt{b^2-4 a c}-b\right ) (a+x (b+c x))^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.071, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ex \right ) ^{m} \left ( Bx+A \right ) \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}}\, 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 (B x + A\right )} \left (e x\right )^{m}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{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{\sqrt{c x^{2} + b x + a}{\left (B x + A\right )} \left (e x\right )^{m}}{c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}}, 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 (e x\right )^{m} \left (A + B x\right )}{\left (a + b x + c x^{2}\right )^{\frac{3}{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 (B x + A\right )} \left (e x\right )^{m}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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