3.2512 \(\int (d+e x)^2 \left (a+b x+c x^2\right )^{5/4} \, dx\)

Optimal. Leaf size=384 \[ \frac{5 \left (b^2-4 a c\right )^{9/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+11 b d)+13 b^2 e^2+44 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{7392 \sqrt{2} c^{17/4} (b+2 c x)}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2} \left (-4 c e (2 a e+11 b d)+13 b^2 e^2+44 c^2 d^2\right )}{3696 c^4}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/4} \left (-4 c e (2 a e+11 b d)+13 b^2 e^2+44 c^2 d^2\right )}{308 c^3}+\frac{13 e \left (a+b x+c x^2\right )^{9/4} (2 c d-b e)}{99 c^2}+\frac{2 e (d+e x) \left (a+b x+c x^2\right )^{9/4}}{11 c} \]

[Out]

(-5*(b^2 - 4*a*c)*(44*c^2*d^2 + 13*b^2*e^2 - 4*c*e*(11*b*d + 2*a*e))*(b + 2*c*x)
*(a + b*x + c*x^2)^(1/4))/(3696*c^4) + ((44*c^2*d^2 + 13*b^2*e^2 - 4*c*e*(11*b*d
+ 2*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(5/4))/(308*c^3) + (13*e*(2*c*d - b*e)*
(a + b*x + c*x^2)^(9/4))/(99*c^2) + (2*e*(d + e*x)*(a + b*x + c*x^2)^(9/4))/(11*
c) + (5*(b^2 - 4*a*c)^(9/4)*(44*c^2*d^2 + 13*b^2*e^2 - 4*c*e*(11*b*d + 2*a*e))*S
qrt[(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])*Ellipti
cF[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2]
)/(7392*Sqrt[2]*c^(17/4)*(b + 2*c*x))

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Rubi [A]  time = 1.13366, antiderivative size = 384, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{5 \left (b^2-4 a c\right )^{9/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+11 b d)+13 b^2 e^2+44 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{7392 \sqrt{2} c^{17/4} (b+2 c x)}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2} \left (-4 c e (2 a e+11 b d)+13 b^2 e^2+44 c^2 d^2\right )}{3696 c^4}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/4} \left (-4 c e (2 a e+11 b d)+13 b^2 e^2+44 c^2 d^2\right )}{308 c^3}+\frac{13 e \left (a+b x+c x^2\right )^{9/4} (2 c d-b e)}{99 c^2}+\frac{2 e (d+e x) \left (a+b x+c x^2\right )^{9/4}}{11 c} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-5*(b^2 - 4*a*c)*(44*c^2*d^2 + 13*b^2*e^2 - 4*c*e*(11*b*d + 2*a*e))*(b + 2*c*x)
*(a + b*x + c*x^2)^(1/4))/(3696*c^4) + ((44*c^2*d^2 + 13*b^2*e^2 - 4*c*e*(11*b*d
+ 2*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(5/4))/(308*c^3) + (13*e*(2*c*d - b*e)*
(a + b*x + c*x^2)^(9/4))/(99*c^2) + (2*e*(d + e*x)*(a + b*x + c*x^2)^(9/4))/(11*
c) + (5*(b^2 - 4*a*c)^(9/4)*(44*c^2*d^2 + 13*b^2*e^2 - 4*c*e*(11*b*d + 2*a*e))*S
qrt[(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])*Ellipti
cF[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2]
)/(7392*Sqrt[2]*c^(17/4)*(b + 2*c*x))

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Rubi in Sympy [A]  time = 87.254, size = 442, normalized size = 1.15 \[ \frac{2 e \left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac{9}{4}}}{11 c} - \frac{13 e \left (b e - 2 c d\right ) \left (a + b x + c x^{2}\right )^{\frac{9}{4}}}{99 c^{2}} + \frac{\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{4}} \left (- 8 a c e^{2} + 13 b^{2} e^{2} - 44 b c d e + 44 c^{2} d^{2}\right )}{308 c^{3}} - \frac{5 \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \sqrt [4]{a + b x + c x^{2}} \left (- 8 a c e^{2} + 13 b^{2} e^{2} - 44 b c d e + 44 c^{2} d^{2}\right )}{3696 c^{4}} + \frac{5 \sqrt{2} \sqrt{- \frac{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}{\left (4 a c - b^{2}\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{9}{4}} \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right ) \left (- 8 a c e^{2} + 13 b^{2} e^{2} - 44 b c d e + 44 c^{2} d^{2}\right ) \sqrt{\left (b + 2 c x\right )^{2}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a + b x + c x^{2}}}{\sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | \frac{1}{2}\right )}{14784 c^{\frac{17}{4}} \left (b + 2 c x\right ) \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*e*(d + e*x)*(a + b*x + c*x**2)**(9/4)/(11*c) - 13*e*(b*e - 2*c*d)*(a + b*x + c
*x**2)**(9/4)/(99*c**2) + (b + 2*c*x)*(a + b*x + c*x**2)**(5/4)*(-8*a*c*e**2 + 1
3*b**2*e**2 - 44*b*c*d*e + 44*c**2*d**2)/(308*c**3) - 5*(b + 2*c*x)*(-4*a*c + b*
*2)*(a + b*x + c*x**2)**(1/4)*(-8*a*c*e**2 + 13*b**2*e**2 - 44*b*c*d*e + 44*c**2
*d**2)/(3696*c**4) + 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)*(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2) +
1)*(-8*a*c*e**2 + 13*b**2*e**2 - 44*b*c*d*e + 44*c**2*d**2)*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)/(14784*c**(17/4)*(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 = 1.47008, size = 416, normalized size = 1.08 \[ \frac{15 \sqrt [4]{2} \left (b^2-4 a c\right )^2 \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{3/4} \left (-4 c e (2 a e+11 b d)+13 b^2 e^2+44 c^2 d^2\right ) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )-4 c (a+x (b+c x)) \left (-16 b c^2 \left (-139 a^2 e^2+8 a c \left (33 d^2+11 d e x+2 e^2 x^2\right )+c^2 x^2 \left (297 d^2+418 d e x+161 e^2 x^2\right )\right )-32 c^3 \left (a^2 e (154 d+15 e x)+4 a c x \left (66 d^2+77 d e x+27 e^2 x^2\right )+c^2 x^3 \left (99 d^2+154 d e x+63 e^2 x^2\right )\right )+4 b^3 c \left (c \left (165 d^2+66 d e x+13 e^2 x^2\right )-342 a e^2\right )+8 b^2 c^2 \left (2 a e (264 d+29 e x)-c x \left (33 d^2+22 d e x+5 e^2 x^2\right )\right )+195 b^5 e^2-6 b^4 c e (110 d+13 e x)\right )}{44352 c^5 (a+x (b+c x))^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*c*(a + x*(b + c*x))*(195*b^5*e^2 - 6*b^4*c*e*(110*d + 13*e*x) + 8*b^2*c^2*(2
*a*e*(264*d + 29*e*x) - c*x*(33*d^2 + 22*d*e*x + 5*e^2*x^2)) + 4*b^3*c*(-342*a*e
^2 + c*(165*d^2 + 66*d*e*x + 13*e^2*x^2)) - 32*c^3*(a^2*e*(154*d + 15*e*x) + 4*a
*c*x*(66*d^2 + 77*d*e*x + 27*e^2*x^2) + c^2*x^3*(99*d^2 + 154*d*e*x + 63*e^2*x^2
)) - 16*b*c^2*(-139*a^2*e^2 + 8*a*c*(33*d^2 + 11*d*e*x + 2*e^2*x^2) + c^2*x^2*(2
97*d^2 + 418*d*e*x + 161*e^2*x^2))) + 15*2^(1/4)*(b^2 - 4*a*c)^2*(44*c^2*d^2 + 1
3*b^2*e^2 - 4*c*e*(11*b*d + 2*a*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])])/(44352*c^5*(a + x*(b + c
*x))^(3/4))

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Maple [F]  time = 0.129, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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