3.2524 \(\int \frac{(d+e x)^3}{\left (a+b x+c x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=307 \[ \frac{\sqrt [4]{b^2-4 a c} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{4 \sqrt{2} c^{13/4} (b+2 c x)}+\frac{e \sqrt [4]{a+b x+c x^2} \left (-2 c e (8 a e+25 b d)+15 b^2 e^2+6 c e x (2 c d-b e)+56 c^2 d^2\right )}{10 c^3}+\frac{2 e (d+e x)^2 \sqrt [4]{a+b x+c x^2}}{5 c} \]

[Out]

(2*e*(d + e*x)^2*(a + b*x + c*x^2)^(1/4))/(5*c) + (e*(56*c^2*d^2 + 15*b^2*e^2 -
2*c*e*(25*b*d + 8*a*e) + 6*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(1/4))/(10*c^3
) + ((b^2 - 4*a*c)^(1/4)*(2*c*d - b*e)*(4*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(b*d + 2*a
*e))*Sqrt[(b + 2*c*x)^2/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sq
rt[b^2 - 4*a*c])^2)]*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*E
llipticF[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)]
, 1/2])/(4*Sqrt[2]*c^(13/4)*(b + 2*c*x))

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Rubi [A]  time = 0.776405, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\sqrt [4]{b^2-4 a c} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{4 \sqrt{2} c^{13/4} (b+2 c x)}+\frac{e \sqrt [4]{a+b x+c x^2} \left (-2 c e (8 a e+25 b d)+15 b^2 e^2+6 c e x (2 c d-b e)+56 c^2 d^2\right )}{10 c^3}+\frac{2 e (d+e x)^2 \sqrt [4]{a+b x+c x^2}}{5 c} \]

Antiderivative was successfully verified.

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

[Out]

(2*e*(d + e*x)^2*(a + b*x + c*x^2)^(1/4))/(5*c) + (e*(56*c^2*d^2 + 15*b^2*e^2 -
2*c*e*(25*b*d + 8*a*e) + 6*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(1/4))/(10*c^3
) + ((b^2 - 4*a*c)^(1/4)*(2*c*d - b*e)*(4*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(b*d + 2*a
*e))*Sqrt[(b + 2*c*x)^2/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sq
rt[b^2 - 4*a*c])^2)]*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*E
llipticF[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)]
, 1/2])/(4*Sqrt[2]*c^(13/4)*(b + 2*c*x))

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Rubi in Sympy [A]  time = 79.5092, size = 369, normalized size = 1.2 \[ \frac{2 e \left (d + e x\right )^{2} \sqrt [4]{a + b x + c x^{2}}}{5 c} + \frac{8 e \sqrt [4]{a + b x + c x^{2}} \left (- 3 a c e^{2} + \frac{45 b^{2} e^{2}}{16} - \frac{75 b c d e}{8} + \frac{21 c^{2} d^{2}}{2} - \frac{9 c e x \left (b e - 2 c d\right )}{8}\right )}{15 c^{3}} - \frac{\sqrt{2} \sqrt{- \frac{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}{\left (4 a c - b^{2}\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )^{2}}} \sqrt [4]{- 4 a c + b^{2}} \left (b e - 2 c d\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right ) \left (- 8 a c e^{2} + 3 b^{2} e^{2} - 4 b c d e + 4 c^{2} d^{2}\right ) \sqrt{\left (b + 2 c x\right )^{2}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a + b x + c x^{2}}}{\sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | \frac{1}{2}\right )}{8 c^{\frac{13}{4}} \left (b + 2 c x\right ) \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*e*(d + e*x)**2*(a + b*x + c*x**2)**(1/4)/(5*c) + 8*e*(a + b*x + c*x**2)**(1/4)
*(-3*a*c*e**2 + 45*b**2*e**2/16 - 75*b*c*d*e/8 + 21*c**2*d**2/2 - 9*c*e*x*(b*e -
2*c*d)/8)/(15*c**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)**(1/4)*(b*e - 2*c*d)*(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4
*a*c + b**2) + 1)*(-8*a*c*e**2 + 3*b**2*e**2 - 4*b*c*d*e + 4*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)/(8*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 = 0.734274, size = 230, normalized size = 0.75 \[ \frac{5 \sqrt [4]{2} \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{3/4} (2 c d-b e) \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right ) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )+4 c e (a+x (b+c x)) \left (-2 c e (8 a e+25 b d+3 b e x)+15 b^2 e^2+4 c^2 \left (15 d^2+5 d e x+e^2 x^2\right )\right )}{40 c^4 (a+x (b+c x))^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(4*c*e*(a + x*(b + c*x))*(15*b^2*e^2 - 2*c*e*(25*b*d + 8*a*e + 3*b*e*x) + 4*c^2*
(15*d^2 + 5*d*e*x + e^2*x^2)) + 5*2^(1/4)*(2*c*d - b*e)*(4*c^2*d^2 + 3*b^2*e^2 -
4*c*e*(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])])/(40*c^4*(a + x*(b + c*x))^(3/4))

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Maple [F]  time = 0.131, size = 0, normalized size = 0. \[ \int{ \left ( ex+d \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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