3.2533 \(\int \frac{d+e x}{\left (a+b x+c x^2\right )^{5/4}} \, dx\)

Optimal. Leaf size=490 \[ \frac{\sqrt{2} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{c^{3/4} \sqrt [4]{b^2-4 a c} (b+2 c x)}-\frac{2 \sqrt{2} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{c^{3/4} \sqrt [4]{b^2-4 a c} (b+2 c x)}+\frac{4 (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e)}{\sqrt{c} \left (b^2-4 a c\right )^{3/2} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}-\frac{4 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.8893, antiderivative size = 490, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\sqrt{2} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{c^{3/4} \sqrt [4]{b^2-4 a c} (b+2 c x)}-\frac{2 \sqrt{2} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{c^{3/4} \sqrt [4]{b^2-4 a c} (b+2 c x)}+\frac{4 (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e)}{\sqrt{c} \left (b^2-4 a c\right )^{3/2} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}-\frac{4 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 77.7654, size = 626, normalized size = 1.28 \[ \frac{4 \left (2 a e - b d + x \left (b e - 2 c d\right )\right )}{\left (- 4 a c + b^{2}\right ) \sqrt [4]{a + b x + c x^{2}}} - \frac{4 \left (b e - 2 c d\right ) \sqrt [4]{a + b x + c x^{2}} \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )} \sqrt{\left (b + 2 c x\right )^{2}}}{\sqrt{c} \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{\frac{3}{2}} \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )} + \frac{2 \sqrt{2} \sqrt{- \frac{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}{\left (4 a c - b^{2}\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )^{2}}} \left (b e - 2 c d\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right ) \sqrt{\left (b + 2 c x\right )^{2}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a + b x + c x^{2}}}{\sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | \frac{1}{2}\right )}{c^{\frac{3}{4}} \left (b + 2 c x\right ) \sqrt [4]{- 4 a c + b^{2}} \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}} - \frac{\sqrt{2} \sqrt{- \frac{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}{\left (4 a c - b^{2}\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )^{2}}} \left (b e - 2 c d\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right ) \sqrt{\left (b + 2 c x\right )^{2}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a + b x + c x^{2}}}{\sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | \frac{1}{2}\right )}{c^{\frac{3}{4}} \left (b + 2 c x\right ) \sqrt [4]{- 4 a c + b^{2}} \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*(2*a*e - b*d + x*(b*e - 2*c*d))/((-4*a*c + b**2)*(a + b*x + c*x**2)**(1/4)) -
4*(b*e - 2*c*d)*(a + b*x + c*x**2)**(1/4)*sqrt(-4*a*c + b**2 + c*(4*a + 4*b*x +
4*c*x**2))*sqrt((b + 2*c*x)**2)/(sqrt(c)*(b + 2*c*x)*(-4*a*c + b**2)**(3/2)*(2*s
qrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2) + 1)) + 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))*(b*e - 2*c*d)*(2*sqrt(c)*sqrt(a + b*x + c
*x**2)/sqrt(-4*a*c + b**2) + 1)*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)/(c**(3/4)*(b + 2
*c*x)*(-4*a*c + b**2)**(1/4)*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*s
qrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2) + 1)**2))*(b*e - 2*c*d)*(2*sqr
t(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2) + 1)*sqrt((b + 2*c*x)**2)*ellipt
ic_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)/(c**(3/4)*(b + 2*c*x)*(-4*a*c + b**2)**(1/4)*sqrt(-4*a*c + b**2 + c*(4*a +
4*b*x + 4*c*x**2)))

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Mathematica [C]  time = 0.351518, size = 167, normalized size = 0.34 \[ -\frac{2 \left (2^{3/4} \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \sqrt [4]{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}} (b e-2 c d) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )+6 c (-2 a e+b (d-e x)+2 c d x)\right )}{3 c \left (b^2-4 a c\right ) \sqrt [4]{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(6*c*(-2*a*e + 2*c*d*x + b*(d - e*x)) + 2^(3/4)*(-2*c*d + b*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)*Hyp
ergeometric2F1[1/4, 3/4, 7/4, (-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/(2*Sqrt[b^2 - 4*a
*c])]))/(3*c*(b^2 - 4*a*c)*(a + x*(b + c*x))^(1/4))

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Maple [F]  time = 0.117, size = 0, normalized size = 0. \[ \int{(ex+d) \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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