3.2509 \(\int \frac{\left (a+b x+c x^2\right )^{3/4}}{d+e x} \, dx\)

Optimal. Leaf size=1209 \[ \text{result too large to display} \]

[Out]

(2*(a + b*x + c*x^2)^(3/4))/(3*e) - ((2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)
^(1/4))/(Sqrt[c]*Sqrt[b^2 - 4*a*c]*e^2*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sq
rt[b^2 - 4*a*c])) + ((-b^2 + 4*a*c)^(1/4)*(c*d^2 - b*d*e + a*e^2)^(3/4)*(-((c*(a
+ b*x + c*x^2))/(b^2 - 4*a*c)))^(1/4)*ArcTan[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*(1 -
(b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4))/(Sqrt[2]*c^(1/4)*(c*d^2 - b*d*e + a*e^2)^(1
/4))])/(c^(1/4)*e^(5/2)*(a + b*x + c*x^2)^(1/4)) - ((-b^2 + 4*a*c)^(1/4)*(c*d^2
- b*d*e + a*e^2)^(3/4)*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(1/4)*ArcTanh[((
-b^2 + 4*a*c)^(1/4)*Sqrt[e]*(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4))/(Sqrt[2]*c^
(1/4)*(c*d^2 - b*d*e + a*e^2)^(1/4))])/(c^(1/4)*e^(5/2)*(a + b*x + c*x^2)^(1/4))
+ ((b^2 - 4*a*c)^(3/4)*(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])/(Sqrt[2]*c^(3/4)*e^2*(b + 2*c*x)) - ((b
^2 - 4*a*c)^(3/4)*(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])/(2*Sqrt[2]*c^(3/4)*e^2*(b + 2*c*x)) - (Sqrt[-
b^2 + 4*a*c]*(2*c*d - b*e)*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[(b + 2*c*x)^2/(b^2 -
4*a*c)]*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(1/4)*EllipticPi[-(Sqrt[-b^2 +
4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 - b*d*e + a*e^2]), ArcSin[(1 - (b + 2*c*x)^2/(b
^2 - 4*a*c))^(1/4)], -1])/(Sqrt[2]*Sqrt[c]*e^3*(b + 2*c*x)*(a + b*x + c*x^2)^(1/
4)) + (Sqrt[-b^2 + 4*a*c]*(2*c*d - b*e)*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[(b + 2*
c*x)^2/(b^2 - 4*a*c)]*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(1/4)*EllipticPi[
(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 - b*d*e + a*e^2]), ArcSin[(1 - (b +
2*c*x)^2/(b^2 - 4*a*c))^(1/4)], -1])/(Sqrt[2]*Sqrt[c]*e^3*(b + 2*c*x)*(a + b*x
+ c*x^2)^(1/4))

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Rubi [A]  time = 6.23576, antiderivative size = 1209, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 18, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.818 \[ -\frac{(2 c d-b e) \sqrt [4]{c x^2+b x+a} (b+2 c x)}{\sqrt{c} \sqrt{b^2-4 a c} e^2 \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right )}+\frac{\sqrt [4]{4 a c-b^2} \left (c d^2-b e d+a e^2\right )^{3/4} \sqrt [4]{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \tan ^{-1}\left (\frac{\sqrt [4]{4 a c-b^2} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{\sqrt [4]{c} e^{5/2} \sqrt [4]{c x^2+b x+a}}-\frac{\sqrt [4]{4 a c-b^2} \left (c d^2-b e d+a e^2\right )^{3/4} \sqrt [4]{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \tanh ^{-1}\left (\frac{\sqrt [4]{4 a c-b^2} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{\sqrt [4]{c} e^{5/2} \sqrt [4]{c x^2+b x+a}}+\frac{2 \left (c x^2+b x+a\right )^{3/4}}{3 e}+\frac{\left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\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 )}{\sqrt{2} c^{3/4} e^2 (b+2 c x)}-\frac{\left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\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 )}{2 \sqrt{2} c^{3/4} e^2 (b+2 c x)}-\frac{\sqrt{4 a c-b^2} (2 c d-b e) \sqrt{c d^2-b e d+a e^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \Pi \left (-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} \sqrt{c} e^3 \sqrt [4]{c x^2+b x+a} (b+2 c x)}+\frac{\sqrt{4 a c-b^2} (2 c d-b e) \sqrt{c d^2-b e d+a e^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \Pi \left (\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} \sqrt{c} e^3 \sqrt [4]{c x^2+b x+a} (b+2 c x)} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*(a + b*x + c*x^2)^(3/4))/(3*e) - ((2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)
^(1/4))/(Sqrt[c]*Sqrt[b^2 - 4*a*c]*e^2*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sq
rt[b^2 - 4*a*c])) + ((-b^2 + 4*a*c)^(1/4)*(c*d^2 - b*d*e + a*e^2)^(3/4)*(-((c*(a
+ b*x + c*x^2))/(b^2 - 4*a*c)))^(1/4)*ArcTan[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*(1 -
(b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4))/(Sqrt[2]*c^(1/4)*(c*d^2 - b*d*e + a*e^2)^(1
/4))])/(c^(1/4)*e^(5/2)*(a + b*x + c*x^2)^(1/4)) - ((-b^2 + 4*a*c)^(1/4)*(c*d^2
- b*d*e + a*e^2)^(3/4)*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(1/4)*ArcTanh[((
-b^2 + 4*a*c)^(1/4)*Sqrt[e]*(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4))/(Sqrt[2]*c^
(1/4)*(c*d^2 - b*d*e + a*e^2)^(1/4))])/(c^(1/4)*e^(5/2)*(a + b*x + c*x^2)^(1/4))
+ ((b^2 - 4*a*c)^(3/4)*(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])/(Sqrt[2]*c^(3/4)*e^2*(b + 2*c*x)) - ((b
^2 - 4*a*c)^(3/4)*(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])/(2*Sqrt[2]*c^(3/4)*e^2*(b + 2*c*x)) - (Sqrt[-
b^2 + 4*a*c]*(2*c*d - b*e)*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[(b + 2*c*x)^2/(b^2 -
4*a*c)]*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(1/4)*EllipticPi[-(Sqrt[-b^2 +
4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 - b*d*e + a*e^2]), ArcSin[(1 - (b + 2*c*x)^2/(b
^2 - 4*a*c))^(1/4)], -1])/(Sqrt[2]*Sqrt[c]*e^3*(b + 2*c*x)*(a + b*x + c*x^2)^(1/
4)) + (Sqrt[-b^2 + 4*a*c]*(2*c*d - b*e)*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[(b + 2*
c*x)^2/(b^2 - 4*a*c)]*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(1/4)*EllipticPi[
(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 - b*d*e + a*e^2]), ArcSin[(1 - (b +
2*c*x)^2/(b^2 - 4*a*c))^(1/4)], -1])/(Sqrt[2]*Sqrt[c]*e^3*(b + 2*c*x)*(a + b*x
+ c*x^2)^(1/4))

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

[Out]

Timed out

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Mathematica [C]  time = 0.533231, size = 180, normalized size = 0.15 \[ \frac{4 \sqrt{2} (a+x (b+c x))^{3/4} F_1\left (-\frac{3}{2};-\frac{3}{4},-\frac{3}{4};-\frac{1}{2};\frac{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right )}{3 e \left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{3/4} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{3/4}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(4*Sqrt[2]*(a + x*(b + c*x))^(3/4)*AppellF1[-3/2, -3/4, -3/4, -1/2, (2*c*d - (b
+ Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*
c*d + 2*c*e*x)])/(3*e*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(3/4)*
((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(3/4))

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Maple [F]  time = 0.154, size = 0, normalized size = 0. \[ \int{\frac{1}{ex+d} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)

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

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]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError