3.2506 \(\int (d+e x)^2 \left (a+b x+c x^2\right )^{3/4} \, dx\)

Optimal. Leaf size=630 \[ -\frac{\sqrt{b^2-4 a c} (b+2 c x) \sqrt [4]{a+b x+c x^2} \left (-4 c e (2 a e+9 b d)+11 b^2 e^2+36 c^2 d^2\right )}{120 c^{7/2} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}-\frac{\left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+9 b d)+11 b^2 e^2+36 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{240 \sqrt{2} c^{15/4} (b+2 c x)}+\frac{\left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+9 b d)+11 b^2 e^2+36 c^2 d^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{120 \sqrt{2} c^{15/4} (b+2 c x)}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/4} \left (-4 c e (2 a e+9 b d)+11 b^2 e^2+36 c^2 d^2\right )}{180 c^3}+\frac{11 e \left (a+b x+c x^2\right )^{7/4} (2 c d-b e)}{63 c^2}+\frac{2 e (d+e x) \left (a+b x+c x^2\right )^{7/4}}{9 c} \]

[Out]

((36*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(9*b*d + 2*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)
^(3/4))/(180*c^3) + (11*e*(2*c*d - b*e)*(a + b*x + c*x^2)^(7/4))/(63*c^2) + (2*e
*(d + e*x)*(a + b*x + c*x^2)^(7/4))/(9*c) - (Sqrt[b^2 - 4*a*c]*(36*c^2*d^2 + 11*
b^2*e^2 - 4*c*e*(9*b*d + 2*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4))/(120*c^(7/
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)*(36*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(9*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*S
qrt[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])/(120*Sqrt[2]*c^(15/4)*
(b + 2*c*x)) - ((b^2 - 4*a*c)^(7/4)*(36*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(9*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])/S
qrt[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])/(240*Sqrt[2]*c^(15/4)*(b + 2*c*x))

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Rubi [A]  time = 1.68925, antiderivative size = 630, 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 \[ -\frac{\sqrt{b^2-4 a c} (b+2 c x) \sqrt [4]{a+b x+c x^2} \left (-4 c e (2 a e+9 b d)+11 b^2 e^2+36 c^2 d^2\right )}{120 c^{7/2} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}-\frac{\left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+9 b d)+11 b^2 e^2+36 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{240 \sqrt{2} c^{15/4} (b+2 c x)}+\frac{\left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+9 b d)+11 b^2 e^2+36 c^2 d^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{120 \sqrt{2} c^{15/4} (b+2 c x)}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/4} \left (-4 c e (2 a e+9 b d)+11 b^2 e^2+36 c^2 d^2\right )}{180 c^3}+\frac{11 e \left (a+b x+c x^2\right )^{7/4} (2 c d-b e)}{63 c^2}+\frac{2 e (d+e x) \left (a+b x+c x^2\right )^{7/4}}{9 c} \]

Warning: Unable to verify antiderivative.

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

[Out]

((36*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(9*b*d + 2*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)
^(3/4))/(180*c^3) + (11*e*(2*c*d - b*e)*(a + b*x + c*x^2)^(7/4))/(63*c^2) + (2*e
*(d + e*x)*(a + b*x + c*x^2)^(7/4))/(9*c) - (Sqrt[b^2 - 4*a*c]*(36*c^2*d^2 + 11*
b^2*e^2 - 4*c*e*(9*b*d + 2*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4))/(120*c^(7/
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)*(36*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(9*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*S
qrt[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])/(120*Sqrt[2]*c^(15/4)*
(b + 2*c*x)) - ((b^2 - 4*a*c)^(7/4)*(36*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(9*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])/S
qrt[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])/(240*Sqrt[2]*c^(15/4)*(b + 2*c*x))

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Rubi in Sympy [A]  time = 159.036, size = 775, normalized size = 1.23 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*e*(d + e*x)*(a + b*x + c*x**2)**(7/4)/(9*c) - 11*e*(b*e - 2*c*d)*(a + b*x + c*
x**2)**(7/4)/(63*c**2) + (b + 2*c*x)*(a + b*x + c*x**2)**(3/4)*(-8*a*c*e**2 + 11
*b**2*e**2 - 36*b*c*d*e + 36*c**2*d**2)/(180*c**3) - sqrt(-4*a*c + b**2)*(a + b*
x + c*x**2)**(1/4)*sqrt(-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c*x**2))*(-8*a*c*e**2
+ 11*b**2*e**2 - 36*b*c*d*e + 36*c**2*d**2)*sqrt((b + 2*c*x)**2)/(120*c**(7/2)*
(b + 2*c*x)*(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2) + 1)) + 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)*(2*s
qrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2) + 1)*(-8*a*c*e**2 + 11*b**2*e*
*2 - 36*b*c*d*e + 36*c**2*d**2)*sqrt((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)/(240*c**(15/4)*(
b + 2*c*x)*sqrt(-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c*x**2))) - 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)*(2*sqrt(c)*sqrt
(a + b*x + c*x**2)/sqrt(-4*a*c + b**2) + 1)*(-8*a*c*e**2 + 11*b**2*e**2 - 36*b*c
*d*e + 36*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)/(480*c**(15/4)*(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 = 0.967716, size = 275, normalized size = 0.44 \[ \frac{4 c (a+x (b+c x)) \left (12 b c \left (c \left (21 d^2+18 d e x+5 e^2 x^2\right )-23 a e^2\right )+8 c^2 \left (3 a e (30 d+7 e x)+c x \left (63 d^2+90 d e x+35 e^2 x^2\right )\right )+77 b^3 e^2-6 b^2 c e (42 d+11 e x)\right )-7\ 2^{3/4} \left (b^2-4 a c\right ) \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \sqrt [4]{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}} \left (-4 c e (2 a e+9 b d)+11 b^2 e^2+36 c^2 d^2\right ) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{5040 c^4 \sqrt [4]{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(4*c*(a + x*(b + c*x))*(77*b^3*e^2 - 6*b^2*c*e*(42*d + 11*e*x) + 12*b*c*(-23*a*e
^2 + c*(21*d^2 + 18*d*e*x + 5*e^2*x^2)) + 8*c^2*(3*a*e*(30*d + 7*e*x) + c*x*(63*
d^2 + 90*d*e*x + 35*e^2*x^2))) - 7*2^(3/4)*(b^2 - 4*a*c)*(36*c^2*d^2 + 11*b^2*e^
2 - 4*c*e*(9*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)*Hypergeometric2F1[1/4, 3/4, 7/4, (-b + Sqrt
[b^2 - 4*a*c] - 2*c*x)/(2*Sqrt[b^2 - 4*a*c])])/(5040*c^4*(a + x*(b + c*x))^(1/4)
)

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Maple [F]  time = 0.123, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x)**2*(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)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError