3.2505 \(\int (d+e x)^3 \left (a+b x+c x^2\right )^{3/4} \, dx\)

Optimal. Leaf size=703 \[ -\frac{\sqrt{b^2-4 a c} (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right )}{80 c^{9/2} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}-\frac{\left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{160 \sqrt{2} c^{19/4} (b+2 c x)}+\frac{\left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{80 \sqrt{2} c^{19/4} (b+2 c x)}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/4} (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right )}{120 c^4}+\frac{e \left (a+b x+c x^2\right )^{7/4} \left (-2 c e (24 a e+121 b d)+55 b^2 e^2+70 c e x (2 c d-b e)+312 c^2 d^2\right )}{462 c^3}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c} \]

[Out]

((2*c*d - b*e)*(12*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))*(b + 2*c*x)*(a +
b*x + c*x^2)^(3/4))/(120*c^4) + (2*e*(d + e*x)^2*(a + b*x + c*x^2)^(7/4))/(11*c
) + (e*(312*c^2*d^2 + 55*b^2*e^2 - 2*c*e*(121*b*d + 24*a*e) + 70*c*e*(2*c*d - b*
e)*x)*(a + b*x + c*x^2)^(7/4))/(462*c^3) - (Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(12*
c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4)
)/(80*c^(9/2)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])) + ((b^2
- 4*a*c)^(7/4)*(2*c*d - b*e)*(12*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*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
cE[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2]
)/(80*Sqrt[2]*c^(19/4)*(b + 2*c*x)) - ((b^2 - 4*a*c)^(7/4)*(2*c*d - b*e)*(12*c^2
*d^2 + 5*b^2*e^2 - 4*c*e*(3*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])/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])/(160*Sqrt[2]*c^(19/4)*(b + 2*c*x))

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Rubi [A]  time = 1.75197, antiderivative size = 703, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ -\frac{\sqrt{b^2-4 a c} (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right )}{80 c^{9/2} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}-\frac{\left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{160 \sqrt{2} c^{19/4} (b+2 c x)}+\frac{\left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{80 \sqrt{2} c^{19/4} (b+2 c x)}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/4} (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right )}{120 c^4}+\frac{e \left (a+b x+c x^2\right )^{7/4} \left (-2 c e (24 a e+121 b d)+55 b^2 e^2+70 c e x (2 c d-b e)+312 c^2 d^2\right )}{462 c^3}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c} \]

Warning: Unable to verify antiderivative.

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

[Out]

((2*c*d - b*e)*(12*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))*(b + 2*c*x)*(a +
b*x + c*x^2)^(3/4))/(120*c^4) + (2*e*(d + e*x)^2*(a + b*x + c*x^2)^(7/4))/(11*c
) + (e*(312*c^2*d^2 + 55*b^2*e^2 - 2*c*e*(121*b*d + 24*a*e) + 70*c*e*(2*c*d - b*
e)*x)*(a + b*x + c*x^2)^(7/4))/(462*c^3) - (Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(12*
c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4)
)/(80*c^(9/2)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])) + ((b^2
- 4*a*c)^(7/4)*(2*c*d - b*e)*(12*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*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
cE[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2]
)/(80*Sqrt[2]*c^(19/4)*(b + 2*c*x)) - ((b^2 - 4*a*c)^(7/4)*(2*c*d - b*e)*(12*c^2
*d^2 + 5*b^2*e^2 - 4*c*e*(3*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])/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])/(160*Sqrt[2]*c^(19/4)*(b + 2*c*x))

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

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]

Timed out

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Mathematica [C]  time = 1.62716, size = 377, normalized size = 0.54 \[ \frac{\frac{4 (a+x (b+c x)) \left (16 c^2 \left (-60 a^2 e^3+3 a c e \left (165 d^2+77 d e x+15 e^2 x^2\right )+c^2 x \left (231 d^3+495 d^2 e x+385 d e^2 x^2+105 e^3 x^3\right )\right )-12 b^2 c e \left (c \left (231 d^2+121 d e x+25 e^2 x^2\right )-143 a e^2\right )+8 b c^2 \left (c \left (231 d^3+297 d^2 e x+165 d e^2 x^2+35 e^3 x^3\right )-3 a e^2 (253 d+47 e x)\right )-385 b^4 e^3+22 b^3 c e^2 (77 d+15 e x)\right )}{77 c^4}+\frac{2^{3/4} \left (b^2-4 a c\right ) \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \sqrt [4]{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}} (b e-2 c d) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right ) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c^5}}{480 \sqrt [4]{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

((4*(a + x*(b + c*x))*(-385*b^4*e^3 + 22*b^3*c*e^2*(77*d + 15*e*x) - 12*b^2*c*e*
(-143*a*e^2 + c*(231*d^2 + 121*d*e*x + 25*e^2*x^2)) + 8*b*c^2*(-3*a*e^2*(253*d +
47*e*x) + c*(231*d^3 + 297*d^2*e*x + 165*d*e^2*x^2 + 35*e^3*x^3)) + 16*c^2*(-60
*a^2*e^3 + 3*a*c*e*(165*d^2 + 77*d*e*x + 15*e^2*x^2) + c^2*x*(231*d^3 + 495*d^2*
e*x + 385*d*e^2*x^2 + 105*e^3*x^3))))/(77*c^4) + (2^(3/4)*(b^2 - 4*a*c)*(-2*c*d
+ b*e)*(12*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*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])^(1/4)*Hypergeometric
2F1[1/4, 3/4, 7/4, (-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/(2*Sqrt[b^2 - 4*a*c])])/c^5)
/(480*(a + x*(b + c*x))^(1/4))

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Maple [F]  time = 0.141, 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. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{3}{4}}{\left (e x + d\right )}^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/4)*(e*x + d)^3,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. \[ \int \left (d + e x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac{3}{4}}\, dx \]

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(-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)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError