Optimal. Leaf size=281 \[ \frac {A (e x)^{m+1} \left (a+b x+c x^2\right )^{3/2} F_1\left (m+1;-\frac {3}{2},-\frac {3}{2};m+2;-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{e (m+1) \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )^{3/2} \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )^{3/2}}+\frac {B (e x)^{m+2} \left (a+b x+c x^2\right )^{3/2} F_1\left (m+2;-\frac {3}{2},-\frac {3}{2};m+3;-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{e^2 (m+2) \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )^{3/2} \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )^{3/2}} \]
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Rubi [A] time = 0.41, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {843, 759, 133} \[ \frac {A (e x)^{m+1} \left (a+b x+c x^2\right )^{3/2} F_1\left (m+1;-\frac {3}{2},-\frac {3}{2};m+2;-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{e (m+1) \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )^{3/2} \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )^{3/2}}+\frac {B (e x)^{m+2} \left (a+b x+c x^2\right )^{3/2} F_1\left (m+2;-\frac {3}{2},-\frac {3}{2};m+3;-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{e^2 (m+2) \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )^{3/2} \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 133
Rule 759
Rule 843
Rubi steps
\begin {align*} \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx &=A \int (e x)^m \left (a+b x+c x^2\right )^{3/2} \, dx+\frac {B \int (e x)^{1+m} \left (a+b x+c x^2\right )^{3/2} \, dx}{e}\\ &=\frac {\left (B \left (a+b x+c x^2\right )^{3/2}\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 )^{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 (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}}+\frac {\left (A \left (a+b x+c x^2\right )^{3/2}\right ) \operatorname {Subst}\left (\int x^m \left (1+\frac {2 c x}{\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^{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 (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}}\\ &=\frac {A (e x)^{1+m} \left (a+b x+c x^2\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 (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}}+\frac {B (e x)^{2+m} \left (a+b x+c x^2\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 (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}}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 405, normalized size = 1.44 \[ \frac {x (e x)^m \sqrt {a+x (b+c x)} \left ((m+1) x \left (\left (m^2+7 m+12\right ) (a B+A b) F_1\left (m+2;-\frac {1}{2},-\frac {1}{2};m+3;-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{\sqrt {b^2-4 a c}-b}\right )+(m+2) x \left ((m+4) (A c+b B) F_1\left (m+3;-\frac {1}{2},-\frac {1}{2};m+4;-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{\sqrt {b^2-4 a c}-b}\right )+B c (m+3) x F_1\left (m+4;-\frac {1}{2},-\frac {1}{2};m+5;-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{\sqrt {b^2-4 a c}-b}\right )\right )\right )+a A \left (m^3+9 m^2+26 m+24\right ) F_1\left (m+1;-\frac {1}{2},-\frac {1}{2};m+2;-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{\sqrt {b^2-4 a c}-b}\right )\right )}{(m+1) (m+2) (m+3) (m+4) \sqrt {\frac {-\sqrt {b^2-4 a c}+b+2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}+b}}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B c x^{3} + {\left (B b + A c\right )} x^{2} + A a + {\left (B a + A b\right )} x\right )} \sqrt {c x^{2} + b x + a} \left (e x\right )^{m}, x\right ) \]
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 \[ \int {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (B x + A\right )} \left (e x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.44, size = 0, normalized size = 0.00 \[ \int \left (B x +A \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (e x \right )^{m}\, dx \]
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 \[ \int {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (B x + A\right )} \left (e x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (e\,x\right )}^m\,\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{m} \left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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