3.1091 \(\int \frac{(e x)^m (A+B x)}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=281 \[ \frac{A (e x)^{m+1} \sqrt{\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1} F_1\left (m+1;\frac{1}{2},\frac{1}{2};m+2;-\frac{2 c x}{b-\sqrt{b^2-4 a c}},-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{e (m+1) \sqrt{a+b x+c x^2}}+\frac{B (e x)^{m+2} \sqrt{\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1} F_1\left (m+2;\frac{1}{2},\frac{1}{2};m+3;-\frac{2 c x}{b-\sqrt{b^2-4 a c}},-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{e^2 (m+2) \sqrt{a+b x+c x^2}} \]

[Out]

(A*(e*x)^(1 + m)*Sqrt[1 + (2*c*x)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x)/(b +
Sqrt[b^2 - 4*a*c])]*AppellF1[1 + m, 1/2, 1/2, 2 + m, (-2*c*x)/(b - Sqrt[b^2 - 4
*a*c]), (-2*c*x)/(b + Sqrt[b^2 - 4*a*c])])/(e*(1 + m)*Sqrt[a + b*x + c*x^2]) + (
B*(e*x)^(2 + m)*Sqrt[1 + (2*c*x)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x)/(b +
Sqrt[b^2 - 4*a*c])]*AppellF1[2 + m, 1/2, 1/2, 3 + m, (-2*c*x)/(b - Sqrt[b^2 - 4*
a*c]), (-2*c*x)/(b + Sqrt[b^2 - 4*a*c])])/(e^2*(2 + m)*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 1.23405, 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 \[ \frac{A (e x)^{m+1} \sqrt{\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1} F_1\left (m+1;\frac{1}{2},\frac{1}{2};m+2;-\frac{2 c x}{b-\sqrt{b^2-4 a c}},-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{e (m+1) \sqrt{a+b x+c x^2}}+\frac{B (e x)^{m+2} \sqrt{\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1} F_1\left (m+2;\frac{1}{2},\frac{1}{2};m+3;-\frac{2 c x}{b-\sqrt{b^2-4 a c}},-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{e^2 (m+2) \sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[((e*x)^m*(A + B*x))/Sqrt[a + b*x + c*x^2],x]

[Out]

(A*(e*x)^(1 + m)*Sqrt[1 + (2*c*x)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x)/(b +
Sqrt[b^2 - 4*a*c])]*AppellF1[1 + m, 1/2, 1/2, 2 + m, (-2*c*x)/(b - Sqrt[b^2 - 4
*a*c]), (-2*c*x)/(b + Sqrt[b^2 - 4*a*c])])/(e*(1 + m)*Sqrt[a + b*x + c*x^2]) + (
B*(e*x)^(2 + m)*Sqrt[1 + (2*c*x)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x)/(b +
Sqrt[b^2 - 4*a*c])]*AppellF1[2 + m, 1/2, 1/2, 3 + m, (-2*c*x)/(b - Sqrt[b^2 - 4*
a*c]), (-2*c*x)/(b + Sqrt[b^2 - 4*a*c])])/(e^2*(2 + m)*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 86.5419, size = 245, normalized size = 0.87 \[ \frac{A \left (e x\right )^{m + 1} \sqrt{\frac{2 c x}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x}{b + \sqrt{- 4 a c + b^{2}}} + 1} \operatorname{appellf_{1}}{\left (m + 1,\frac{1}{2},\frac{1}{2},m + 2,- \frac{2 c x}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{e \left (m + 1\right ) \sqrt{a + b x + c x^{2}}} + \frac{B \left (e x\right )^{m + 2} \sqrt{\frac{2 c x}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x}{b + \sqrt{- 4 a c + b^{2}}} + 1} \operatorname{appellf_{1}}{\left (m + 2,\frac{1}{2},\frac{1}{2},m + 3,- \frac{2 c x}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{e^{2} \left (m + 2\right ) \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m*(B*x+A)/(c*x**2+b*x+a)**(1/2),x)

[Out]

A*(e*x)**(m + 1)*sqrt(2*c*x/(b - sqrt(-4*a*c + b**2)) + 1)*sqrt(2*c*x/(b + sqrt(
-4*a*c + b**2)) + 1)*appellf1(m + 1, 1/2, 1/2, m + 2, -2*c*x/(b - sqrt(-4*a*c +
b**2)), -2*c*x/(b + sqrt(-4*a*c + b**2)))/(e*(m + 1)*sqrt(a + b*x + c*x**2)) + B
*(e*x)**(m + 2)*sqrt(2*c*x/(b - sqrt(-4*a*c + b**2)) + 1)*sqrt(2*c*x/(b + sqrt(-
4*a*c + b**2)) + 1)*appellf1(m + 2, 1/2, 1/2, m + 3, -2*c*x/(b - sqrt(-4*a*c + b
**2)), -2*c*x/(b + sqrt(-4*a*c + b**2)))/(e**2*(m + 2)*sqrt(a + b*x + c*x**2))

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Mathematica [B]  time = 4.32821, size = 614, normalized size = 2.19 \[ \frac{a x (e x)^m \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (-\frac{A (m+2)^2 F_1\left (m+1;\frac{1}{2},\frac{1}{2};m+2;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )}{(m+1) \left (-4 a (m+2) F_1\left (m+1;\frac{1}{2},\frac{1}{2};m+2;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )+x \left (\sqrt{b^2-4 a c}+b\right ) F_1\left (m+2;\frac{1}{2},\frac{3}{2};m+3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )+x \left (b-\sqrt{b^2-4 a c}\right ) F_1\left (m+2;\frac{3}{2},\frac{1}{2};m+3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )\right )}-\frac{B (m+3) x F_1\left (m+2;\frac{1}{2},\frac{1}{2};m+3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )}{-4 a (m+3) F_1\left (m+2;\frac{1}{2},\frac{1}{2};m+3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )+x \left (\sqrt{b^2-4 a c}+b\right ) F_1\left (m+3;\frac{1}{2},\frac{3}{2};m+4;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )+x \left (b-\sqrt{b^2-4 a c}\right ) F_1\left (m+3;\frac{3}{2},\frac{1}{2};m+4;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )}\right )}{c (m+2) (a+x (b+c x))^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((e*x)^m*(A + B*x))/Sqrt[a + b*x + c*x^2],x]

[Out]

(a*x*(e*x)^m*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)*(-(
(A*(2 + m)^2*AppellF1[1 + m, 1/2, 1/2, 2 + m, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]),
(2*c*x)/(-b + Sqrt[b^2 - 4*a*c])])/((1 + m)*(-4*a*(2 + m)*AppellF1[1 + m, 1/2, 1
/2, 2 + m, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])] +
(b + Sqrt[b^2 - 4*a*c])*x*AppellF1[2 + m, 1/2, 3/2, 3 + m, (-2*c*x)/(b + Sqrt[b
^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4*a*c])*x*Appe
llF1[2 + m, 3/2, 1/2, 3 + m, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqr
t[b^2 - 4*a*c])]))) - (B*(3 + m)*x*AppellF1[2 + m, 1/2, 1/2, 3 + m, (-2*c*x)/(b
+ Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])])/(-4*a*(3 + m)*AppellF1[
2 + m, 1/2, 1/2, 3 + m, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2
- 4*a*c])] + (b + Sqrt[b^2 - 4*a*c])*x*AppellF1[3 + m, 1/2, 3/2, 4 + m, (-2*c*x
)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4
*a*c])*x*AppellF1[3 + m, 3/2, 1/2, 4 + m, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c
*x)/(-b + Sqrt[b^2 - 4*a*c])])))/(c*(2 + m)*(a + x*(b + c*x))^(3/2))

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Maple [F]  time = 0.116, size = 0, normalized size = 0. \[ \int{ \left ( ex \right ) ^{m} \left ( Bx+A \right ){\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^m*(B*x+A)/(c*x^2+b*x+a)^(1/2),x)

[Out]

int((e*x)^m*(B*x+A)/(c*x^2+b*x+a)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} \left (e x\right )^{m}}{\sqrt{c x^{2} + b x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x)^m/sqrt(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

integrate((B*x + A)*(e*x)^m/sqrt(c*x^2 + b*x + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x + A\right )} \left (e x\right )^{m}}{\sqrt{c x^{2} + b x + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x)^m/sqrt(c*x^2 + b*x + a),x, algorithm="fricas")

[Out]

integral((B*x + A)*(e*x)^m/sqrt(c*x^2 + b*x + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (e x\right )^{m} \left (A + B x\right )}{\sqrt{a + b x + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**m*(B*x+A)/(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral((e*x)**m*(A + B*x)/sqrt(a + b*x + c*x**2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} \left (e x\right )^{m}}{\sqrt{c x^{2} + b x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x)^m/sqrt(c*x^2 + b*x + a),x, algorithm="giac")

[Out]

integrate((B*x + A)*(e*x)^m/sqrt(c*x^2 + b*x + a), x)