3.2513 \(\int (d+e x) \left (a+b x+c x^2\right )^{5/4} \, dx\)

Optimal. Leaf size=285 \[ \frac{5 \left (b^2-4 a c\right )^{9/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{336 \sqrt{2} c^{13/4} (b+2 c x)}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e)}{168 c^3}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/4} (2 c d-b e)}{14 c^2}+\frac{2 e \left (a+b x+c x^2\right )^{9/4}}{9 c} \]

[Out]

(-5*(b^2 - 4*a*c)*(2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4))/(168*c^3) +
((2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(5/4))/(14*c^2) + (2*e*(a + b*x +
c*x^2)^(9/4))/(9*c) + (5*(b^2 - 4*a*c)^(9/4)*(2*c*d - b*e)*Sqrt[(b + 2*c*x)^2/((
b^2 - 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])^2)]*(1 +
(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*EllipticF[2*ArcTan[(Sqrt[2]
*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(336*Sqrt[2]*c^(13
/4)*(b + 2*c*x))

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Rubi [A]  time = 0.534573, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{5 \left (b^2-4 a c\right )^{9/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{336 \sqrt{2} c^{13/4} (b+2 c x)}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e)}{168 c^3}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/4} (2 c d-b e)}{14 c^2}+\frac{2 e \left (a+b x+c x^2\right )^{9/4}}{9 c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)*(a + b*x + c*x^2)^(5/4),x]

[Out]

(-5*(b^2 - 4*a*c)*(2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4))/(168*c^3) +
((2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(5/4))/(14*c^2) + (2*e*(a + b*x +
c*x^2)^(9/4))/(9*c) + (5*(b^2 - 4*a*c)^(9/4)*(2*c*d - b*e)*Sqrt[(b + 2*c*x)^2/((
b^2 - 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])^2)]*(1 +
(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*EllipticF[2*ArcTan[(Sqrt[2]
*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(336*Sqrt[2]*c^(13
/4)*(b + 2*c*x))

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Rubi in Sympy [A]  time = 44.4328, size = 330, normalized size = 1.16 \[ \frac{2 e \left (a + b x + c x^{2}\right )^{\frac{9}{4}}}{9 c} - \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{4}}}{14 c^{2}} + \frac{5 \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \left (b e - 2 c d\right ) \sqrt [4]{a + b x + c x^{2}}}{168 c^{3}} - \frac{5 \sqrt{2} \sqrt{- \frac{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}{\left (4 a c - b^{2}\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{9}{4}} \left (b e - 2 c d\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right ) \sqrt{\left (b + 2 c x\right )^{2}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a + b x + c x^{2}}}{\sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | \frac{1}{2}\right )}{672 c^{\frac{13}{4}} \left (b + 2 c x\right ) \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)*(c*x**2+b*x+a)**(5/4),x)

[Out]

2*e*(a + b*x + c*x**2)**(9/4)/(9*c) - (b + 2*c*x)*(b*e - 2*c*d)*(a + b*x + c*x**
2)**(5/4)/(14*c**2) + 5*(b + 2*c*x)*(-4*a*c + b**2)*(b*e - 2*c*d)*(a + b*x + c*x
**2)**(1/4)/(168*c**3) - 5*sqrt(2)*sqrt(-(-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c*x
**2))/((4*a*c - b**2)*(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2) + 1)
**2))*(-4*a*c + b**2)**(9/4)*(b*e - 2*c*d)*(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqr
t(-4*a*c + b**2) + 1)*sqrt((b + 2*c*x)**2)*elliptic_f(2*atan(sqrt(2)*c**(1/4)*(a
+ b*x + c*x**2)**(1/4)/(-4*a*c + b**2)**(1/4)), 1/2)/(672*c**(13/4)*(b + 2*c*x)
*sqrt(-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c*x**2)))

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Mathematica [C]  time = 0.812595, size = 270, normalized size = 0.95 \[ \frac{-4 c (a+x (b+c x)) \left (-16 c^2 \left (7 a^2 e+2 a c x (12 d+7 e x)+c^2 x^3 (9 d+7 e x)\right )-4 b^2 c (c x (3 d+e x)-24 a e)-8 b c^2 \left (4 a (6 d+e x)+c x^2 (27 d+19 e x)\right )-15 b^4 e+6 b^3 c (5 d+e x)\right )-15 \sqrt [4]{2} \left (b^2-4 a c\right )^2 \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{3/4} (b e-2 c d) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{2016 c^4 (a+x (b+c x))^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)*(a + b*x + c*x^2)^(5/4),x]

[Out]

(-4*c*(a + x*(b + c*x))*(-15*b^4*e + 6*b^3*c*(5*d + e*x) - 4*b^2*c*(-24*a*e + c*
x*(3*d + e*x)) - 16*c^2*(7*a^2*e + c^2*x^3*(9*d + 7*e*x) + 2*a*c*x*(12*d + 7*e*x
)) - 8*b*c^2*(4*a*(6*d + e*x) + c*x^2*(27*d + 19*e*x))) - 15*2^(1/4)*(b^2 - 4*a*
c)^2*(-2*c*d + b*e)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*((b + Sqrt[b^2 - 4*a*c] + 2*
c*x)/Sqrt[b^2 - 4*a*c])^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, (-b + Sqrt[b^2 -
4*a*c] - 2*c*x)/(2*Sqrt[b^2 - 4*a*c])])/(2016*c^4*(a + x*(b + c*x))^(3/4))

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Maple [F]  time = 0.123, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(c*x^2+b*x+a)^(5/4),x)

[Out]

int((e*x+d)*(c*x^2+b*x+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{5}{4}}{\left (e x + d\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/4)*(e*x + d),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(5/4)*(e*x + d), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c e x^{3} +{\left (c d + b e\right )} x^{2} + a d +{\left (b d + a e\right )} x\right )}{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/4)*(e*x + d),x, algorithm="fricas")

[Out]

integral((c*e*x^3 + (c*d + b*e)*x^2 + a*d + (b*d + a*e)*x)*(c*x^2 + b*x + a)^(1/
4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)*(c*x**2+b*x+a)**(5/4),x)

[Out]

Integral((d + e*x)*(a + b*x + c*x**2)**(5/4), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{5}{4}}{\left (e x + d\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/4)*(e*x + d),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(5/4)*(e*x + d), x)