3.2515 \(\int \frac{\left (a+b x+c x^2\right )^{5/4}}{d+e x} \, dx\)

Optimal. Leaf size=1014 \[ \text{result too large to display} \]

[Out]

((12*c^2*d^2 + b^2*e^2 - 2*c*e*(7*b*d - 6*a*e) - 2*c*e*(2*c*d - b*e)*x)*(a + b*x
+ c*x^2)^(1/4))/(6*c*e^3) + (2*(a + b*x + c*x^2)^(5/4))/(5*e) - ((-b^2 + 4*a*c)
^(3/4)*(c*d^2 - b*d*e + a*e^2)^(5/4)*(-((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))])/(c^(3/4)*e^(7/2)*(a + b*x +
c*x^2)^(3/4)) - ((-b^2 + 4*a*c)^(3/4)*(c*d^2 - b*d*e + a*e^2)^(5/4)*(-((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)
)])/(c^(3/4)*e^(7/2)*(a + b*x + c*x^2)^(3/4)) - ((b^2 - 4*a*c)^(1/4)*(2*c*d - b*
e)*(12*c^2*d^2 - b^2*e^2 - 4*c*e*(3*b*d - 4*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])/(12*Sqrt[2]*c^(5/4)*e^4*(b
+ 2*c*x)) - ((b^2 - 4*a*c)*(2*c*d - b*e)*(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)))^(3/4)*EllipticPi[-(S
qrt[-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]*c*e^4*(b + 2*c*x)*(a + b*x + c*x^2)
^(3/4)) - ((b^2 - 4*a*c)*(2*c*d - b*e)*(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)))^(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])/(Sqrt[2]*c*e^4*(b + 2*c*x)*(a + b*x + c*x^2)^(3
/4))

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Rubi [A]  time = 7.07784, antiderivative size = 1014, 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{\left (4 a c-b^2\right )^{3/4} \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 ) \left (c d^2-b e d+a e^2\right )^{5/4}}{c^{3/4} e^{7/2} \left (c x^2+b x+a\right )^{3/4}}-\frac{\left (4 a c-b^2\right )^{3/4} \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 ) \left (c d^2-b e d+a e^2\right )^{5/4}}{c^{3/4} e^{7/2} \left (c x^2+b x+a\right )^{3/4}}-\frac{\left (b^2-4 a c\right ) (2 c d-b e) \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 ) \left (c d^2-b e d+a e^2\right )}{\sqrt{2} c e^4 (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}-\frac{\left (b^2-4 a c\right ) (2 c d-b e) \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 ) \left (c d^2-b e d+a e^2\right )}{\sqrt{2} c e^4 (b+2 c x) \left (c x^2+b x+a\right )^{3/4}}+\frac{2 \left (c x^2+b x+a\right )^{5/4}}{5 e}-\frac{\sqrt [4]{b^2-4 a c} (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 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 )}{12 \sqrt{2} c^{5/4} e^4 (b+2 c x)}+\frac{\left (12 c^2 d^2+b^2 e^2-2 c e (7 b d-6 a e)-2 c e (2 c d-b e) x\right ) \sqrt [4]{c x^2+b x+a}}{6 c e^3} \]

Warning: Unable to verify antiderivative.

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

[Out]

((12*c^2*d^2 + b^2*e^2 - 2*c*e*(7*b*d - 6*a*e) - 2*c*e*(2*c*d - b*e)*x)*(a + b*x
+ c*x^2)^(1/4))/(6*c*e^3) + (2*(a + b*x + c*x^2)^(5/4))/(5*e) - ((-b^2 + 4*a*c)
^(3/4)*(c*d^2 - b*d*e + a*e^2)^(5/4)*(-((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))])/(c^(3/4)*e^(7/2)*(a + b*x +
c*x^2)^(3/4)) - ((-b^2 + 4*a*c)^(3/4)*(c*d^2 - b*d*e + a*e^2)^(5/4)*(-((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)
)])/(c^(3/4)*e^(7/2)*(a + b*x + c*x^2)^(3/4)) - ((b^2 - 4*a*c)^(1/4)*(2*c*d - b*
e)*(12*c^2*d^2 - b^2*e^2 - 4*c*e*(3*b*d - 4*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])/(12*Sqrt[2]*c^(5/4)*e^4*(b
+ 2*c*x)) - ((b^2 - 4*a*c)*(2*c*d - b*e)*(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)))^(3/4)*EllipticPi[-(S
qrt[-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]*c*e^4*(b + 2*c*x)*(a + b*x + c*x^2)
^(3/4)) - ((b^2 - 4*a*c)*(2*c*d - b*e)*(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)))^(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])/(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),x)

[Out]

Timed out

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Mathematica [A]  time = 2.26943, size = 0, normalized size = 0. \[ \int \frac{\left (a+b x+c x^2\right )^{5/4}}{d+e x} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.163, size = 0, normalized size = 0. \[ \int{\frac{1}{ex+d} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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