3.2532 \(\int \frac{(d+e x)^2}{\left (a+b x+c x^2\right )^{5/4}} \, dx\)

Optimal. Leaf size=594 \[ \frac{2 (b+2 c x) \sqrt [4]{a+b x+c x^2} \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )}{c^{3/2} \left (b^2-4 a c\right )^{3/2} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}+\frac{\sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt{2} c^{7/4} \sqrt [4]{b^2-4 a c} (b+2 c x)}-\frac{\sqrt{2} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{c^{7/4} \sqrt [4]{b^2-4 a c} (b+2 c x)}+\frac{4 e \left (a+b x+c x^2\right )^{3/4} (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac{4 (d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}} \]

[Out]

(-4*(d + e*x)*(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*e*(2*c*d - b*e)*(a + b*x + c*x^2)^(3/4))/(c*(b^2 - 4*a*c)) + (2*(4*c
^2*d^2 + 3*b^2*e^2 - 4*c*e*(b*d + 2*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4))/(
c^(3/2)*(b^2 - 4*a*c)^(3/2)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*
a*c])) - (Sqrt[2]*(4*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(b*d + 2*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])*EllipticE[2*ArcTan[(Sq
rt[2]*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(c^(7/4)*(b^2
- 4*a*c)^(1/4)*(b + 2*c*x)) + ((4*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(b*d + 2*a*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])
/(Sqrt[2]*c^(7/4)*(b^2 - 4*a*c)^(1/4)*(b + 2*c*x))

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Rubi [A]  time = 1.53841, antiderivative size = 594, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{2 (b+2 c x) \sqrt [4]{a+b x+c x^2} \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )}{c^{3/2} \left (b^2-4 a c\right )^{3/2} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}+\frac{\sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt{2} c^{7/4} \sqrt [4]{b^2-4 a c} (b+2 c x)}-\frac{\sqrt{2} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{c^{7/4} \sqrt [4]{b^2-4 a c} (b+2 c x)}+\frac{4 e \left (a+b x+c x^2\right )^{3/4} (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac{4 (d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-4*(d + e*x)*(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*e*(2*c*d - b*e)*(a + b*x + c*x^2)^(3/4))/(c*(b^2 - 4*a*c)) + (2*(4*c
^2*d^2 + 3*b^2*e^2 - 4*c*e*(b*d + 2*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4))/(
c^(3/2)*(b^2 - 4*a*c)^(3/2)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*
a*c])) - (Sqrt[2]*(4*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(b*d + 2*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])*EllipticE[2*ArcTan[(Sq
rt[2]*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(c^(7/4)*(b^2
- 4*a*c)^(1/4)*(b + 2*c*x)) + ((4*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(b*d + 2*a*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])
/(Sqrt[2]*c^(7/4)*(b^2 - 4*a*c)^(1/4)*(b + 2*c*x))

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Rubi in Sympy [A]  time = 144.854, size = 743, normalized size = 1.25 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*(d + e*x)*(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*e*(b*e - 2*c*d)*(a + b*x + c*x**2)**(3/4)/(c*(-4*a*c + b**2)) + (a +
b*x + c*x**2)**(1/4)*sqrt(-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c*x**2))*(-32*a*c*
e**2 + 12*b**2*e**2 - 16*b*c*d*e + 16*c**2*d**2)*sqrt((b + 2*c*x)**2)/(2*c**(3/2
)*(b + 2*c*x)*(-4*a*c + b**2)**(3/2)*(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a
*c + b**2) + 1)) - 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))*(2
*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2) + 1)*(-32*a*c*e**2 + 12*b**2
*e**2 - 16*b*c*d*e + 16*c**2*d**2)*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)/(4*c**(7/4)*(
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*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2) + 1)**2))*(2*sqrt(c)*sqrt
(a + b*x + c*x**2)/sqrt(-4*a*c + b**2) + 1)*(-32*a*c*e**2 + 12*b**2*e**2 - 16*b*
c*d*e + 16*c**2*d**2)*sqrt((b + 2*c*x)**2)*elliptic_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)/(8*c**(7/4)*(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 = 0.564361, size = 221, normalized size = 0.37 \[ \frac{24 \left (a b e^2-2 a c e (2 d+e x)+b^2 e^2 x+b c d (d-2 e x)+2 c^2 d^2 x\right )-\frac{2\ 2^{3/4} \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \sqrt [4]{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}} \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right ) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c}}{6 c \left (4 a c-b^2\right ) \sqrt [4]{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(24*(a*b*e^2 + 2*c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 2*e*x) - 2*a*c*e*(2*d + e*x)
) - (2*2^(3/4)*(4*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(b*d + 2*a*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)*Hypergeom
etric2F1[1/4, 3/4, 7/4, (-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/(2*Sqrt[b^2 - 4*a*c])])
/c)/(6*c*(-b^2 + 4*a*c)*(a + x*(b + c*x))^(1/4))

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Maple [F]  time = 0.13, size = 0, normalized size = 0. \[ \int{ \left ( ex+d \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((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. \[ \int \frac{{\left (e x + d\right )}^{2}}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((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. \[ \int \frac{{\left (e x + d\right )}^{2}}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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