3.2545 \(\int (d+e x)^m \left (a+b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=189 \[ \frac{\left (a+b x+c x^2\right )^{5/2} (d+e x)^{m+1} F_1\left (m+1;-\frac{5}{2},-\frac{5}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e (m+1) \left (1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}\right )^{5/2} \left (1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )^{5/2}} \]

[Out]

((d + e*x)^(1 + m)*(a + b*x + c*x^2)^(5/2)*AppellF1[1 + m, -5/2, -5/2, 2 + m, (2
*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e), (2*c*(d + e*x))/(2*c*d - (b +
Sqrt[b^2 - 4*a*c])*e)])/(e*(1 + m)*(1 - (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2
- 4*a*c])*e))^(5/2)*(1 - (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e))^(5
/2))

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Rubi [A]  time = 0.844824, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\left (a+b x+c x^2\right )^{5/2} (d+e x)^{m+1} F_1\left (m+1;-\frac{5}{2},-\frac{5}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e (m+1) \left (1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}\right )^{5/2} \left (1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^m*(a + b*x + c*x^2)^(5/2),x]

[Out]

((d + e*x)^(1 + m)*(a + b*x + c*x^2)^(5/2)*AppellF1[1 + m, -5/2, -5/2, 2 + m, (2
*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e), (2*c*(d + e*x))/(2*c*d - (b +
Sqrt[b^2 - 4*a*c])*e)])/(e*(1 + m)*(1 - (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2
- 4*a*c])*e))^(5/2)*(1 - (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e))^(5
/2))

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Rubi in Sympy [A]  time = 36.8125, size = 175, normalized size = 0.93 \[ \frac{\left (d + e x\right )^{m + 1} \left (a + b x + c x^{2}\right )^{\frac{5}{2}} \operatorname{appellf_{1}}{\left (m + 1,- \frac{5}{2},- \frac{5}{2},m + 2,\frac{c \left (- 2 d - 2 e x\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}},\frac{c \left (2 d + 2 e x\right )}{2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )} \right )}}{e \left (m + 1\right ) \left (\frac{c \left (- 2 d - 2 e x\right )}{2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )} + 1\right )^{\frac{5}{2}} \left (\frac{c \left (2 d + 2 e x\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}} + 1\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**m*(c*x**2+b*x+a)**(5/2),x)

[Out]

(d + e*x)**(m + 1)*(a + b*x + c*x**2)**(5/2)*appellf1(m + 1, -5/2, -5/2, m + 2,
c*(-2*d - 2*e*x)/(b*e - 2*c*d - e*sqrt(-4*a*c + b**2)), c*(2*d + 2*e*x)/(2*c*d -
e*(b + sqrt(-4*a*c + b**2))))/(e*(m + 1)*(c*(-2*d - 2*e*x)/(2*c*d - e*(b + sqrt
(-4*a*c + b**2))) + 1)**(5/2)*(c*(2*d + 2*e*x)/(b*e - 2*c*d - e*sqrt(-4*a*c + b*
*2)) + 1)**(5/2))

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Mathematica [A]  time = 0.2266, size = 0, normalized size = 0. \[ \int (d+e x)^m \left (a+b x+c x^2\right )^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(d + e*x)^m*(a + b*x + c*x^2)^(5/2),x]

[Out]

Integrate[(d + e*x)^m*(a + b*x + c*x^2)^(5/2), x]

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Maple [F]  time = 0.125, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{m} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m*(c*x^2+b*x+a)^(5/2),x)

[Out]

int((e*x+d)^m*(c*x^2+b*x+a)^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}{\left (e x + d\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)*(e*x + d)^m,x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)*(e*x + d)^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{c x^{2} + b x + a}{\left (e x + d\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)*(e*x + d)^m,x, algorithm="fricas")

[Out]

integral((c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*sqrt(c*x^2 +
b*x + a)*(e*x + d)^m, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**m*(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}{\left (e x + d\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)*(e*x + d)^m,x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)*(e*x + d)^m, x)