3.2507 \(\int (d+e x) \left (a+b x+c x^2\right )^{3/4} \, dx\)

Optimal. Leaf size=510 \[ -\frac{3 \sqrt{b^2-4 a c} (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e)}{20 c^{5/2} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}-\frac{3 \left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{40 \sqrt{2} c^{11/4} (b+2 c x)}+\frac{3 \left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{20 \sqrt{2} c^{11/4} (b+2 c x)}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/4} (2 c d-b e)}{10 c^2}+\frac{2 e \left (a+b x+c x^2\right )^{7/4}}{7 c} \]

[Out]

((2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/4))/(10*c^2) + (2*e*(a + b*x + c
*x^2)^(7/4))/(7*c) - (3*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c
*x^2)^(1/4))/(20*c^(5/2)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c
])) + (3*(b^2 - 4*a*c)^(7/4)*(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])*EllipticE[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*
x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(20*Sqrt[2]*c^(11/4)*(b + 2*c*x))
- (3*(b^2 - 4*a*c)^(7/4)*(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])/(40*Sqrt[2]*c^(11/4)*(b + 2*c*x))

_______________________________________________________________________________________

Rubi [A]  time = 0.943562, antiderivative size = 510, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{3 \sqrt{b^2-4 a c} (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e)}{20 c^{5/2} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}-\frac{3 \left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{40 \sqrt{2} c^{11/4} (b+2 c x)}+\frac{3 \left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{20 \sqrt{2} c^{11/4} (b+2 c x)}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/4} (2 c d-b e)}{10 c^2}+\frac{2 e \left (a+b x+c x^2\right )^{7/4}}{7 c} \]

Warning: Unable to verify antiderivative.

[In]  Int[(d + e*x)*(a + b*x + c*x^2)^(3/4),x]

[Out]

((2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/4))/(10*c^2) + (2*e*(a + b*x + c
*x^2)^(7/4))/(7*c) - (3*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c
*x^2)^(1/4))/(20*c^(5/2)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c
])) + (3*(b^2 - 4*a*c)^(7/4)*(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])*EllipticE[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*
x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(20*Sqrt[2]*c^(11/4)*(b + 2*c*x))
- (3*(b^2 - 4*a*c)^(7/4)*(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])/(40*Sqrt[2]*c^(11/4)*(b + 2*c*x))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 92.1657, size = 643, normalized size = 1.26 \[ \frac{2 e \left (a + b x + c x^{2}\right )^{\frac{7}{4}}}{7 c} - \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{4}}}{10 c^{2}} + \frac{3 \sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right ) \sqrt [4]{a + b x + c x^{2}} \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )} \sqrt{\left (b + 2 c x\right )^{2}}}{20 c^{\frac{5}{2}} \left (b + 2 c x\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )} - \frac{3 \sqrt{2} \sqrt{- \frac{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}{\left (4 a c - b^{2}\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} \left (b e - 2 c d\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right ) \sqrt{\left (b + 2 c x\right )^{2}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a + b x + c x^{2}}}{\sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | \frac{1}{2}\right )}{40 c^{\frac{11}{4}} \left (b + 2 c x\right ) \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}} + \frac{3 \sqrt{2} \sqrt{- \frac{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}{\left (4 a c - b^{2}\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} \left (b e - 2 c d\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right ) \sqrt{\left (b + 2 c x\right )^{2}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a + b x + c x^{2}}}{\sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | \frac{1}{2}\right )}{80 c^{\frac{11}{4}} \left (b + 2 c x\right ) \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)*(c*x**2+b*x+a)**(3/4),x)

[Out]

2*e*(a + b*x + c*x**2)**(7/4)/(7*c) - (b + 2*c*x)*(b*e - 2*c*d)*(a + b*x + c*x**
2)**(3/4)/(10*c**2) + 3*sqrt(-4*a*c + b**2)*(b*e - 2*c*d)*(a + b*x + c*x**2)**(1
/4)*sqrt(-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c*x**2))*sqrt((b + 2*c*x)**2)/(20*c*
*(5/2)*(b + 2*c*x)*(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2) + 1)) -
3*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)**(
7/4)*(b*e - 2*c*d)*(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2) + 1)*sq
rt((b + 2*c*x)**2)*elliptic_e(2*atan(sqrt(2)*c**(1/4)*(a + b*x + c*x**2)**(1/4)/
(-4*a*c + b**2)**(1/4)), 1/2)/(40*c**(11/4)*(b + 2*c*x)*sqrt(-4*a*c + b**2 + c*(
4*a + 4*b*x + 4*c*x**2))) + 3*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)**(7/4)*(b*e - 2*c*d)*(2*sqrt(c)*sqrt(a + b*x + c*x**2)/
sqrt(-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)/(80*c**(11/4)*(b + 2*c*
x)*sqrt(-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c*x**2)))

_______________________________________________________________________________________

Mathematica [C]  time = 0.566912, size = 195, normalized size = 0.38 \[ \frac{7\ 2^{3/4} \left (b^2-4 a c\right ) \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \sqrt [4]{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}} (b e-2 c d) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )+4 c (a+x (b+c x)) \left (4 c (5 a e+c x (7 d+5 e x))-7 b^2 e+2 b c (7 d+3 e x)\right )}{280 c^3 \sqrt [4]{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)*(a + b*x + c*x^2)^(3/4),x]

[Out]

(4*c*(a + x*(b + c*x))*(-7*b^2*e + 2*b*c*(7*d + 3*e*x) + 4*c*(5*a*e + c*x*(7*d +
5*e*x))) + 7*2^(3/4)*(b^2 - 4*a*c)*(-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])^(1/4)*Hypergeometric2F1[1
/4, 3/4, 7/4, (-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/(2*Sqrt[b^2 - 4*a*c])])/(280*c^3*
(a + x*(b + c*x))^(1/4))

_______________________________________________________________________________________

Maple [F]  time = 0.119, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(c*x^2+b*x+a)^(3/4),x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/4)*(e*x + d),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/4)*(e*x + d),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)*(c*x**2+b*x+a)**(3/4),x)

[Out]

Integral((d + e*x)*(a + b*x + c*x**2)**(3/4), x)

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/4)*(e*x + d),x, algorithm="giac")

[Out]

Exception raised: TypeError