3.1089 \(\int (e x)^m (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 1.24179, 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 \[ \frac{A (e x)^{m+1} \left (a+b x+c x^2\right )^{3/2} F_1\left (m+1;-\frac{3}{2},-\frac{3}{2};m+2;-\frac{2 c x}{b-\sqrt{b^2-4 a c}},-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{e (m+1) \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )^{3/2} \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )^{3/2}}+\frac{B (e x)^{m+2} \left (a+b x+c x^2\right )^{3/2} F_1\left (m+2;-\frac{3}{2},-\frac{3}{2};m+3;-\frac{2 c x}{b-\sqrt{b^2-4 a c}},-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{e^2 (m+2) \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )^{3/2} \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^m*(A + B*x)*(a + b*x + c*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 85.0992, size = 252, normalized size = 0.9 \[ \frac{A \left (e x\right )^{m + 1} \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \operatorname{appellf_{1}}{\left (m + 1,- \frac{3}{2},- \frac{3}{2},m + 2,- \frac{2 c x}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{e \left (m + 1\right ) \left (\frac{2 c x}{b - \sqrt{- 4 a c + b^{2}}} + 1\right )^{\frac{3}{2}} \left (\frac{2 c x}{b + \sqrt{- 4 a c + b^{2}}} + 1\right )^{\frac{3}{2}}} + \frac{B \left (e x\right )^{m + 2} \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \operatorname{appellf_{1}}{\left (m + 2,- \frac{3}{2},- \frac{3}{2},m + 3,- \frac{2 c x}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{e^{2} \left (m + 2\right ) \left (\frac{2 c x}{b - \sqrt{- 4 a c + b^{2}}} + 1\right )^{\frac{3}{2}} \left (\frac{2 c x}{b + \sqrt{- 4 a c + b^{2}}} + 1\right )^{\frac{3}{2}}} \]

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)**(3/2),x)

[Out]

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

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Mathematica [B]  time = 6.10162, size = 2271, normalized size = 8.08 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(e*x)^m*(A + B*x)*(a + b*x + c*x^2)^(3/2),x]

[Out]

(a*A*(b - Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*(2 + m)*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 + x*(b + c*x))*AppellF
1[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])])/(4*c^2*(1 + m)*(a + b*x + c*x^2)^(3/2)*(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, 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[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])])) + (A*b*(b - Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 -
4*a*c])*(3 + m)*x^2*(e*x)^m*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*(b + Sqrt[b^2 - 4*a
*c] + 2*c*x)*(a + x*(b + c*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*c^2*(2 + m)*(a + b*x +
c*x^2)^(3/2)*(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*Ap
pellF1[3 + m, -1/2, 1/2, 4 + m, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b +
Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4*a*c])*x*AppellF1[3 + m, 1/2, -1/2, 4 + m
, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])])) + (a*B*(
b - Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*(3 + m)*x^2*(e*x)^m*(b - Sqrt[b^2
- 4*a*c] + 2*c*x)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)*(a + x*(b + c*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*c^2*(2 + m)*(a + b*x + c*x^2)^(3/2)*(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, 1/2, 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, 1/2, -1/2, 4 + m, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x
)/(-b + Sqrt[b^2 - 4*a*c])])) + (b*B*(b - Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a
*c])*(4 + m)*x^3*(e*x)^m*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*(b + Sqrt[b^2 - 4*a*c]
+ 2*c*x)*(a + x*(b + c*x))*AppellF1[3 + m, -1/2, -1/2, 4 + m, (-2*c*x)/(b + Sqrt
[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])])/(4*c^2*(3 + m)*(a + b*x + c*x
^2)^(3/2)*(4*a*(4 + m)*AppellF1[3 + m, -1/2, -1/2, 4 + m, (-2*c*x)/(b + Sqrt[b^2
- 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])] + (b + Sqrt[b^2 - 4*a*c])*x*Appell
F1[4 + m, -1/2, 1/2, 5 + 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[4 + m, 1/2, -1/2, 5 + m, (-
2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])])) + (A*(b - Sq
rt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*(4 + m)*x^3*(e*x)^m*(b - Sqrt[b^2 - 4*a
*c] + 2*c*x)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)*(a + x*(b + c*x))*AppellF1[3 + m, -
1/2, -1/2, 4 + m, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a
*c])])/(4*c*(3 + m)*(a + b*x + c*x^2)^(3/2)*(4*a*(4 + m)*AppellF1[3 + m, -1/2, -
1/2, 4 + m, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])]
+ (b + Sqrt[b^2 - 4*a*c])*x*AppellF1[4 + m, -1/2, 1/2, 5 + 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*Ap
pellF1[4 + m, 1/2, -1/2, 5 + m, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b +
Sqrt[b^2 - 4*a*c])])) + (B*(b - Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*(5 +
m)*x^4*(e*x)^m*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)*(
a + x*(b + c*x))*AppellF1[4 + m, -1/2, -1/2, 5 + m, (-2*c*x)/(b + Sqrt[b^2 - 4*a
*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])])/(4*c*(4 + m)*(a + b*x + c*x^2)^(3/2)*(4
*a*(5 + m)*AppellF1[4 + m, -1/2, -1/2, 5 + 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[5 + m, -1
/2, 1/2, 6 + 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[5 + m, 1/2, -1/2, 6 + m, (-2*c*x)/(b +
Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])]))

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Maple [F]  time = 0.115, size = 0, normalized size = 0. \[ \int \left ( ex \right ) ^{m} \left ( Bx+A \right ) \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^m*(B*x+A)*(c*x^2+b*x+a)^(3/2),x)

[Out]

int((e*x)^m*(B*x+A)*(c*x^2+b*x+a)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}{\left (B x + A\right )} \left (e x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)*(B*x + A)*(e*x)^m,x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(3/2)*(B*x + A)*(e*x)^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (B c x^{3} +{\left (B b + A c\right )} x^{2} + A a +{\left (B a + A b\right )} x\right )} \sqrt{c x^{2} + b x + a} \left (e x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)*(B*x + A)*(e*x)^m,x, algorithm="fricas")

[Out]

integral((B*c*x^3 + (B*b + A*c)*x^2 + A*a + (B*a + A*b)*x)*sqrt(c*x^2 + b*x + a)
*(e*x)^m, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (e x\right )^{m} \left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**m*(B*x+A)*(c*x**2+b*x+a)**(3/2),x)

[Out]

Integral((e*x)**m*(A + B*x)*(a + b*x + c*x**2)**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}{\left (B x + A\right )} \left (e x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)*(B*x + A)*(e*x)^m,x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(3/2)*(B*x + A)*(e*x)^m, x)