3.2536 \(\int \frac{1}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/4}} \, dx\)

Optimal. Leaf size=1485 \[ \text{result too large to display} \]

[Out]

(-4*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)*(d + e*x)*(a + b*x + c*x^2)^(1/4)) - (e*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e
*(2*b*d + 3*a*e))*(a + b*x + c*x^2)^(3/4))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2
)^2*(d + e*x)) + (Sqrt[c]*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*(b + 2
*c*x)*(a + b*x + c*x^2)^(1/4))/((b^2 - 4*a*c)^(3/2)*(c*d^2 - b*d*e + a*e^2)^2*(1
+ (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])) + (5*(-b^2 + 4*a*c)^(1/
4)*e^(3/2)*(2*c*d - b*e)*(-((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))])/(4*c^(1/4)*(c*d^2 - b*d*e + a*e^2)^(9/4)
*(a + b*x + c*x^2)^(1/4)) - (5*(-b^2 + 4*a*c)^(1/4)*e^(3/2)*(2*c*d - b*e)*(-((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))])/(4*c^(1/4)*(c*d^2 - b*d*e + a*e^2)^(9/4)*(a + b*x + c*x^2)^(1/4)) - (c
^(1/4)*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*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])/(Sqrt[2]*(b^2 - 4*a*c)
^(1/4)*(c*d^2 - b*d*e + a*e^2)^2*(b + 2*c*x)) + (c^(1/4)*(8*c^2*d^2 + 5*b^2*e^2
- 4*c*e*(2*b*d + 3*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])/(2*Sqrt[2]*(b^2 - 4*a*c)^(1/4)*(c*d^2 - b*d*e + a*e^
2)^2*(b + 2*c*x)) - (5*Sqrt[-b^2 + 4*a*c]*e*(2*c*d - b*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])/(4*Sqrt[2]*Sqrt[c]*(c*d^2 - b*d*e + a*e^2)^(5/2)*
(b + 2*c*x)*(a + b*x + c*x^2)^(1/4)) + (5*Sqrt[-b^2 + 4*a*c]*e*(2*c*d - b*e)^2*S
qrt[(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]), ArcSi
n[(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4)], -1])/(4*Sqrt[2]*Sqrt[c]*(c*d^2 - b*d
*e + a*e^2)^(5/2)*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4))

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Rubi [A]  time = 8.60092, antiderivative size = 1485, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 19, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.864 \[ -\frac{5 \sqrt{4 a c-b^2} e \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 ) (2 c d-b e)^2}{4 \sqrt{2} \sqrt{c} \left (c d^2-b e d+a e^2\right )^{5/2} (b+2 c x) \sqrt [4]{c x^2+b x+a}}+\frac{5 \sqrt{4 a c-b^2} e \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 ) (2 c d-b e)^2}{4 \sqrt{2} \sqrt{c} \left (c d^2-b e d+a e^2\right )^{5/2} (b+2 c x) \sqrt [4]{c x^2+b x+a}}+\frac{5 \sqrt [4]{4 a c-b^2} e^{3/2} \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 ) (2 c d-b e)}{4 \sqrt [4]{c} \left (c d^2-b e d+a e^2\right )^{9/4} \sqrt [4]{c x^2+b x+a}}-\frac{5 \sqrt [4]{4 a c-b^2} e^{3/2} \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 ) (2 c d-b e)}{4 \sqrt [4]{c} \left (c d^2-b e d+a e^2\right )^{9/4} \sqrt [4]{c x^2+b x+a}}-\frac{\sqrt [4]{c} \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \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} \sqrt [4]{b^2-4 a c} \left (c d^2-b e d+a e^2\right )^2 (b+2 c x)}+\frac{\sqrt [4]{c} \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \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} \sqrt [4]{b^2-4 a c} \left (c d^2-b e d+a e^2\right )^2 (b+2 c x)}-\frac{e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \left (c x^2+b x+a\right )^{3/4}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )^2 (d+e x)}-\frac{4 \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) (d+e x) \sqrt [4]{c x^2+b x+a}}+\frac{\sqrt{c} \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) (b+2 c x) \sqrt [4]{c x^2+b x+a}}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b e d+a e^2\right )^2 \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right )} \]

Warning: Unable to verify antiderivative.

[In]  Int[1/((d + e*x)^2*(a + b*x + c*x^2)^(5/4)),x]

[Out]

(-4*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)*(d + e*x)*(a + b*x + c*x^2)^(1/4)) - (e*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e
*(2*b*d + 3*a*e))*(a + b*x + c*x^2)^(3/4))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2
)^2*(d + e*x)) + (Sqrt[c]*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*(b + 2
*c*x)*(a + b*x + c*x^2)^(1/4))/((b^2 - 4*a*c)^(3/2)*(c*d^2 - b*d*e + a*e^2)^2*(1
+ (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])) + (5*(-b^2 + 4*a*c)^(1/
4)*e^(3/2)*(2*c*d - b*e)*(-((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))])/(4*c^(1/4)*(c*d^2 - b*d*e + a*e^2)^(9/4)
*(a + b*x + c*x^2)^(1/4)) - (5*(-b^2 + 4*a*c)^(1/4)*e^(3/2)*(2*c*d - b*e)*(-((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))])/(4*c^(1/4)*(c*d^2 - b*d*e + a*e^2)^(9/4)*(a + b*x + c*x^2)^(1/4)) - (c
^(1/4)*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*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])/(Sqrt[2]*(b^2 - 4*a*c)
^(1/4)*(c*d^2 - b*d*e + a*e^2)^2*(b + 2*c*x)) + (c^(1/4)*(8*c^2*d^2 + 5*b^2*e^2
- 4*c*e*(2*b*d + 3*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])/(2*Sqrt[2]*(b^2 - 4*a*c)^(1/4)*(c*d^2 - b*d*e + a*e^
2)^2*(b + 2*c*x)) - (5*Sqrt[-b^2 + 4*a*c]*e*(2*c*d - b*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])/(4*Sqrt[2]*Sqrt[c]*(c*d^2 - b*d*e + a*e^2)^(5/2)*
(b + 2*c*x)*(a + b*x + c*x^2)^(1/4)) + (5*Sqrt[-b^2 + 4*a*c]*e*(2*c*d - b*e)^2*S
qrt[(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]), ArcSi
n[(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4)], -1])/(4*Sqrt[2]*Sqrt[c]*(c*d^2 - b*d
*e + a*e^2)^(5/2)*(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(1/(e*x+d)**2/(c*x**2+b*x+a)**(5/4),x)

[Out]

Timed out

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Mathematica [C]  time = 1.4153, size = 187, normalized size = 0.13 \[ -\frac{\left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{5/4} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{5/4} F_1\left (\frac{7}{2};\frac{5}{4},\frac{5}{4};\frac{9}{2};\frac{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right )}{14 \sqrt{2} e (d+e x) (a+x (b+c x))^{5/4}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((d + e*x)^2*(a + b*x + c*x^2)^(5/4)),x]

[Out]

-(((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(5/4)*((e*(b + Sqrt[b^2 -
4*a*c] + 2*c*x))/(c*(d + e*x)))^(5/4)*AppellF1[7/2, 5/4, 5/4, 9/2, (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)])/(14*Sqrt[2]*e*(d + e*x)*(a + x*(b + c*x))^(5/4))

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Maple [F]  time = 0.136, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( ex+d \right ) ^{2}} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^2/(c*x^2+b*x+a)^(5/4),x)

[Out]

int(1/(e*x+d)^2/(c*x^2+b*x+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{4}}{\left (e x + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^(5/4)*(e*x + d)^2),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + b*x + a)^(5/4)*(e*x + d)^2), 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(1/((c*x^2 + b*x + a)^(5/4)*(e*x + d)^2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**2/(c*x**2+b*x+a)**(5/4),x)

[Out]

Integral(1/((d + e*x)**2*(a + b*x + c*x**2)**(5/4)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{4}}{\left (e x + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^(5/4)*(e*x + d)^2),x, algorithm="giac")

[Out]

integrate(1/((c*x^2 + b*x + a)^(5/4)*(e*x + d)^2), x)