3.2516 \(\int \frac{\left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=975 \[ \frac{5 \left (b^2-4 a c\right ) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \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} c e^4 (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}+\frac{5 \left (b^2-4 a c\right ) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \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} c e^4 (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}+\frac{5 \left (4 a c-b^2\right )^{3/4} \sqrt [4]{c d^2-b e d+a e^2} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \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 c^{3/4} e^{7/2} \left (c x^2+b x+a\right )^{3/4}}+\frac{5 \left (4 a c-b^2\right )^{3/4} \sqrt [4]{c d^2-b e d+a e^2} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \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 c^{3/4} e^{7/2} \left (c x^2+b x+a\right )^{3/4}}-\frac{\left (c x^2+b x+a\right )^{5/4}}{e (d+e x)}+\frac{5 \sqrt [4]{b^2-4 a c} \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-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 )}{6 \sqrt{2} \sqrt [4]{c} e^4 (b+2 c x)}-\frac{5 (3 c d-2 b e-c e x) \sqrt [4]{c x^2+b x+a}}{3 e^3} \]

[Out]

(-5*(3*c*d - 2*b*e - c*e*x)*(a + b*x + c*x^2)^(1/4))/(3*e^3) - (a + b*x + c*x^2)
^(5/4)/(e*(d + e*x)) + (5*(-b^2 + 4*a*c)^(3/4)*(2*c*d - b*e)*(c*d^2 - b*d*e + a*
e^2)^(1/4)*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(3/4)*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^(3/4)*e^(7/2)*(a + b*x + c*x^2)^(3/4)) + (5*(-b^2
+ 4*a*c)^(3/4)*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^(1/4)*(-((c*(a + b*x + c*x
^2))/(b^2 - 4*a*c)))^(3/4)*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
^(3/4)*e^(7/2)*(a + b*x + c*x^2)^(3/4)) + (5*(b^2 - 4*a*c)^(1/4)*(6*c^2*d^2 + b^
2*e^2 - 2*c*e*(3*b*d - a*e))*Sqrt[(b + 2*c*x)^2/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*S
qrt[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])/(6*Sqrt[2]*c^(1/4)*e^4*(b + 2*c*x)) + (5*(b^2 -
4*a*c)*(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)))^(3/4)*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*Sqr
t[2]*c*e^4*(b + 2*c*x)*(a + b*x + c*x^2)^(3/4)) + (5*(b^2 - 4*a*c)*(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)))^(3
/4)*EllipticPi[(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 - b*d*e + a*e^2]), A
rcSin[(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4)], -1])/(4*Sqrt[2]*c*e^4*(b + 2*c*x
)*(a + b*x + c*x^2)^(3/4))

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Rubi [A]  time = 6.31263, antiderivative size = 975, 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{5 \left (b^2-4 a c\right ) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \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} c e^4 (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}+\frac{5 \left (b^2-4 a c\right ) \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \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} c e^4 (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}+\frac{5 \left (4 a c-b^2\right )^{3/4} \sqrt [4]{c d^2-b e d+a e^2} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \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 c^{3/4} e^{7/2} \left (c x^2+b x+a\right )^{3/4}}+\frac{5 \left (4 a c-b^2\right )^{3/4} \sqrt [4]{c d^2-b e d+a e^2} \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/4} \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 c^{3/4} e^{7/2} \left (c x^2+b x+a\right )^{3/4}}-\frac{\left (c x^2+b x+a\right )^{5/4}}{e (d+e x)}+\frac{5 \sqrt [4]{b^2-4 a c} \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-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 )}{6 \sqrt{2} \sqrt [4]{c} e^4 (b+2 c x)}-\frac{5 (3 c d-2 b e-c e x) \sqrt [4]{c x^2+b x+a}}{3 e^3} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-5*(3*c*d - 2*b*e - c*e*x)*(a + b*x + c*x^2)^(1/4))/(3*e^3) - (a + b*x + c*x^2)
^(5/4)/(e*(d + e*x)) + (5*(-b^2 + 4*a*c)^(3/4)*(2*c*d - b*e)*(c*d^2 - b*d*e + a*
e^2)^(1/4)*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(3/4)*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^(3/4)*e^(7/2)*(a + b*x + c*x^2)^(3/4)) + (5*(-b^2
+ 4*a*c)^(3/4)*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^(1/4)*(-((c*(a + b*x + c*x
^2))/(b^2 - 4*a*c)))^(3/4)*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
^(3/4)*e^(7/2)*(a + b*x + c*x^2)^(3/4)) + (5*(b^2 - 4*a*c)^(1/4)*(6*c^2*d^2 + b^
2*e^2 - 2*c*e*(3*b*d - a*e))*Sqrt[(b + 2*c*x)^2/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*S
qrt[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])/(6*Sqrt[2]*c^(1/4)*e^4*(b + 2*c*x)) + (5*(b^2 -
4*a*c)*(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)))^(3/4)*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*Sqr
t[2]*c*e^4*(b + 2*c*x)*(a + b*x + c*x^2)^(3/4)) + (5*(b^2 - 4*a*c)*(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)))^(3
/4)*EllipticPi[(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 - b*d*e + a*e^2]), A
rcSin[(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4)], -1])/(4*Sqrt[2]*c*e^4*(b + 2*c*x
)*(a + b*x + c*x^2)^(3/4))

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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)**(5/4)/(e*x+d)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 1.76648, size = 0, normalized size = 0. \[ \int \frac{\left (a+b x+c x^2\right )^{5/4}}{(d+e x)^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.219, 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((c*x^2+b*x+a)^(5/4)/(e*x+d)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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Sympy [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)**(5/4)/(e*x+d)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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