Optimal. Leaf size=281 \[ \frac{A (e x)^{m+1} \left (a+b x+c x^2\right )^{5/2} F_1\left (m+1;-\frac{5}{2},-\frac{5}{2};m+2;-\frac{2 c x}{b-\sqrt{b^2-4 a c}},-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{e (m+1) \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )^{5/2} \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )^{5/2}}+\frac{B (e x)^{m+2} \left (a+b x+c x^2\right )^{5/2} F_1\left (m+2;-\frac{5}{2},-\frac{5}{2};m+3;-\frac{2 c x}{b-\sqrt{b^2-4 a c}},-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{e^2 (m+2) \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )^{5/2} \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )^{5/2}} \]
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Rubi [A] time = 0.548072, 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} \[ \frac{A (e x)^{m+1} \left (a+b x+c x^2\right )^{5/2} F_1\left (m+1;-\frac{5}{2},-\frac{5}{2};m+2;-\frac{2 c x}{b-\sqrt{b^2-4 a c}},-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{e (m+1) \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )^{5/2} \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )^{5/2}}+\frac{B (e x)^{m+2} \left (a+b x+c x^2\right )^{5/2} F_1\left (m+2;-\frac{5}{2},-\frac{5}{2};m+3;-\frac{2 c x}{b-\sqrt{b^2-4 a c}},-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{e^2 (m+2) \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )^{5/2} \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 843
Rule 759
Rule 133
Rubi steps
\begin{align*} \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx &=A \int (e x)^m \left (a+b x+c x^2\right )^{5/2} \, dx+\frac{B \int (e x)^{1+m} \left (a+b x+c x^2\right )^{5/2} \, dx}{e}\\ &=\frac{\left (B \left (a+b x+c x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int x^{1+m} \left (1+\frac{2 c x}{\left (b-\sqrt{b^2-4 a c}\right ) e}\right )^{5/2} \left (1+\frac{2 c x}{\left (b+\sqrt{b^2-4 a c}\right ) e}\right )^{5/2} \, dx,x,e x\right )}{e^2 \left (1+\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )^{5/2} \left (1+\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )^{5/2}}+\frac{\left (A \left (a+b x+c x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int x^m \left (1+\frac{2 c x}{\left (b-\sqrt{b^2-4 a c}\right ) e}\right )^{5/2} \left (1+\frac{2 c x}{\left (b+\sqrt{b^2-4 a c}\right ) e}\right )^{5/2} \, dx,x,e x\right )}{e \left (1+\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )^{5/2} \left (1+\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )^{5/2}}\\ &=\frac{A (e x)^{1+m} \left (a+b x+c x^2\right )^{5/2} F_1\left (1+m;-\frac{5}{2},-\frac{5}{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 (1+\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )^{5/2} \left (1+\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )^{5/2}}+\frac{B (e x)^{2+m} \left (a+b x+c x^2\right )^{5/2} F_1\left (2+m;-\frac{5}{2},-\frac{5}{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 (1+\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )^{5/2} \left (1+\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )^{5/2}}\\ \end{align*}
Mathematica [B] time = 1.4887, size = 618, normalized size = 2.2 \[ \frac{x (e x)^m \sqrt{a+x (b+c x)} \left (a^2 A \left (m^5+20 m^4+155 m^3+580 m^2+1044 m+720\right ) F_1\left (m+1;-\frac{1}{2},-\frac{1}{2};m+2;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )+(m+1) x \left (a \left (m^4+18 m^3+119 m^2+342 m+360\right ) (a B+2 A b) F_1\left (m+2;-\frac{1}{2},-\frac{1}{2};m+3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )+(m+2) x \left (\left (m^3+15 m^2+74 m+120\right ) \left (A \left (2 a c+b^2\right )+2 a b B\right ) F_1\left (m+3;-\frac{1}{2},-\frac{1}{2};m+4;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )+(m+3) x \left (\left (m^2+11 m+30\right ) \left (2 a B c+2 A b c+b^2 B\right ) F_1\left (m+4;-\frac{1}{2},-\frac{1}{2};m+5;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )+c (m+4) x \left ((m+6) (A c+2 b B) F_1\left (m+5;-\frac{1}{2},-\frac{1}{2};m+6;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )+B c (m+5) x F_1\left (m+6;-\frac{1}{2},-\frac{1}{2};m+7;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )\right )\right )\right )\right )\right )}{(m+1) (m+2) (m+3) (m+4) (m+5) (m+6) \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}+b}}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.066, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( Bx+A \right ) \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{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{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}{\left (B x + A\right )} \left (e x\right )^{m}\,{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 ({\left (B c^{2} x^{5} +{\left (2 \, B b c + A c^{2}\right )} x^{4} +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{3} + A a^{2} +{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} +{\left (B a^{2} + 2 \, A a b\right )} x\right )} \sqrt{c x^{2} + b x + a} \left (e x\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}{\left (B x + A\right )} \left (e x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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