3.1388 \(\int \frac{1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=325 \[ \frac{112 c^2 \sqrt{a+b x+c x^2}}{d \left (b^2-4 a c\right )^3 \sqrt{b d+2 c d x}}+\frac{56 c \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{d^{3/2} \left (b^2-4 a c\right )^{9/4} \sqrt{a+b x+c x^2}}-\frac{56 c \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{d^{3/2} \left (b^2-4 a c\right )^{9/4} \sqrt{a+b x+c x^2}}+\frac{28 c}{3 d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}-\frac{2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \sqrt{b d+2 c d x}} \]

[Out]

-2/(3*(b^2 - 4*a*c)*d*Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2)^(3/2)) + (28*c)/(3*(
b^2 - 4*a*c)^2*d*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2]) + (112*c^2*Sqrt[a +
b*x + c*x^2])/((b^2 - 4*a*c)^3*d*Sqrt[b*d + 2*c*d*x]) - (56*c*Sqrt[-((c*(a + b*x
+ c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(
1/4)*Sqrt[d])], -1])/((b^2 - 4*a*c)^(9/4)*d^(3/2)*Sqrt[a + b*x + c*x^2]) + (56*c
*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[b*d + 2*c*d*
x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/((b^2 - 4*a*c)^(9/4)*d^(3/2)*Sqrt[a + b*
x + c*x^2])

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Rubi [A]  time = 0.956258, antiderivative size = 325, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{112 c^2 \sqrt{a+b x+c x^2}}{d \left (b^2-4 a c\right )^3 \sqrt{b d+2 c d x}}+\frac{56 c \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{d^{3/2} \left (b^2-4 a c\right )^{9/4} \sqrt{a+b x+c x^2}}-\frac{56 c \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{d^{3/2} \left (b^2-4 a c\right )^{9/4} \sqrt{a+b x+c x^2}}+\frac{28 c}{3 d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}-\frac{2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \sqrt{b d+2 c d x}} \]

Antiderivative was successfully verified.

[In]  Int[1/((b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(5/2)),x]

[Out]

-2/(3*(b^2 - 4*a*c)*d*Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2)^(3/2)) + (28*c)/(3*(
b^2 - 4*a*c)^2*d*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2]) + (112*c^2*Sqrt[a +
b*x + c*x^2])/((b^2 - 4*a*c)^3*d*Sqrt[b*d + 2*c*d*x]) - (56*c*Sqrt[-((c*(a + b*x
+ c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(
1/4)*Sqrt[d])], -1])/((b^2 - 4*a*c)^(9/4)*d^(3/2)*Sqrt[a + b*x + c*x^2]) + (56*c
*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[b*d + 2*c*d*
x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/((b^2 - 4*a*c)^(9/4)*d^(3/2)*Sqrt[a + b*
x + c*x^2])

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Rubi in Sympy [A]  time = 169.63, size = 314, normalized size = 0.97 \[ \frac{112 c^{2} \sqrt{a + b x + c x^{2}}}{d \left (- 4 a c + b^{2}\right )^{3} \sqrt{b d + 2 c d x}} + \frac{28 c}{3 d \left (- 4 a c + b^{2}\right )^{2} \sqrt{b d + 2 c d x} \sqrt{a + b x + c x^{2}}} - \frac{56 c \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{d^{\frac{3}{2}} \left (- 4 a c + b^{2}\right )^{\frac{9}{4}} \sqrt{a + b x + c x^{2}}} + \frac{56 c \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{d^{\frac{3}{2}} \left (- 4 a c + b^{2}\right )^{\frac{9}{4}} \sqrt{a + b x + c x^{2}}} - \frac{2}{3 d \left (- 4 a c + b^{2}\right ) \sqrt{b d + 2 c d x} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*c*d*x+b*d)**(3/2)/(c*x**2+b*x+a)**(5/2),x)

[Out]

112*c**2*sqrt(a + b*x + c*x**2)/(d*(-4*a*c + b**2)**3*sqrt(b*d + 2*c*d*x)) + 28*
c/(3*d*(-4*a*c + b**2)**2*sqrt(b*d + 2*c*d*x)*sqrt(a + b*x + c*x**2)) - 56*c*sqr
t(c*(a + b*x + c*x**2)/(4*a*c - b**2))*elliptic_e(asin(sqrt(b*d + 2*c*d*x)/(sqrt
(d)*(-4*a*c + b**2)**(1/4))), -1)/(d**(3/2)*(-4*a*c + b**2)**(9/4)*sqrt(a + b*x
+ c*x**2)) + 56*c*sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*elliptic_f(asin(sqrt
(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(d**(3/2)*(-4*a*c + b**2)
**(9/4)*sqrt(a + b*x + c*x**2)) - 2/(3*d*(-4*a*c + b**2)*sqrt(b*d + 2*c*d*x)*(a
+ b*x + c*x**2)**(3/2))

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Mathematica [C]  time = 2.26339, size = 270, normalized size = 0.83 \[ \frac{-\frac{2 (b+2 c x) (a+x (b+c x)) \left (-8 c^2 \left (12 a^2+35 a c x^2+21 c^2 x^4\right )-2 b^2 c \left (11 a+91 c x^2\right )-56 b c^2 x \left (5 a+6 c x^2\right )+b^4-14 b^3 c x\right )}{3 \left (b^2-4 a c\right )^3}+\frac{56 i c \left (-\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )^{3/2} (a+x (b+c x))^2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{\left (b^2-4 a c\right )^{3/2}}}{(a+x (b+c x))^{5/2} (d (b+2 c x))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(5/2)),x]

[Out]

((-2*(b + 2*c*x)*(a + x*(b + c*x))*(b^4 - 14*b^3*c*x - 56*b*c^2*x*(5*a + 6*c*x^2
) - 2*b^2*c*(11*a + 91*c*x^2) - 8*c^2*(12*a^2 + 35*a*c*x^2 + 21*c^2*x^4)))/(3*(b
^2 - 4*a*c)^3) + ((56*I)*c*(-((b + 2*c*x)/Sqrt[b^2 - 4*a*c]))^(3/2)*(a + x*(b +
c*x))^2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*(EllipticE[I*ArcSinh[Sqrt[-((
b + 2*c*x)/Sqrt[b^2 - 4*a*c])]], -1] - EllipticF[I*ArcSinh[Sqrt[-((b + 2*c*x)/Sq
rt[b^2 - 4*a*c])]], -1]))/(b^2 - 4*a*c)^(3/2))/((d*(b + 2*c*x))^(3/2)*(a + x*(b
+ c*x))^(5/2))

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Maple [B]  time = 0.046, size = 877, normalized size = 2.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*c*d*x+b*d)^(3/2)/(c*x^2+b*x+a)^(5/2),x)

[Out]

2/3*(168*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2
^(1/2),2^(1/2))*x^2*a*c^3*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2
)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b
^2)^(1/2))^(1/2)-42*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/
2))^(1/2)*2^(1/2),2^(1/2))*x^2*b^2*c^2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2
)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/
2))/(-4*a*c+b^2)^(1/2))^(1/2)+168*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-
4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*x*a*b*c^2*((b+2*c*x+(-4*a*c+b^2)^(1/2))
/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*
a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)-42*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2
)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*x*b^3*c*((b+2*c*x+(-4*a*c+b^
2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2
*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)+168*((b+2*c*x+(-4*a*c+b^2)^(1
/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+
(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^
2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*a^2*c^2-42*((b+2*c*x+(-4*a*
c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((
-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticE(1/2*((b+2*c*x+(
-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*a*b^2*c-168*c^4*x^
4-336*b*c^3*x^3-280*x^2*a*c^3-182*x^2*b^2*c^2-280*x*a*b*c^2-14*b^3*c*x-96*a^2*c^
2-22*a*c*b^2+b^4)*(d*(2*c*x+b))^(1/2)/d^2/(4*a*c-b^2)^3/(2*c*x+b)/(c*x^2+b*x+a)^
(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(3/2)*(c*x^2 + b*x + a)^(5/2)),x, algorithm="maxima")

[Out]

integrate(1/((2*c*d*x + b*d)^(3/2)*(c*x^2 + b*x + a)^(5/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (2 \, c^{3} d x^{5} + 5 \, b c^{2} d x^{4} + 4 \,{\left (b^{2} c + a c^{2}\right )} d x^{3} + a^{2} b d +{\left (b^{3} + 6 \, a b c\right )} d x^{2} + 2 \,{\left (a b^{2} + a^{2} c\right )} d x\right )} \sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(3/2)*(c*x^2 + b*x + a)^(5/2)),x, algorithm="fricas")

[Out]

integral(1/((2*c^3*d*x^5 + 5*b*c^2*d*x^4 + 4*(b^2*c + a*c^2)*d*x^3 + a^2*b*d + (
b^3 + 6*a*b*c)*d*x^2 + 2*(a*b^2 + a^2*c)*d*x)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b
*x + a)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d \left (b + 2 c x\right )\right )^{\frac{3}{2}} \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2*c*d*x+b*d)**(3/2)/(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral(1/((d*(b + 2*c*x))**(3/2)*(a + b*x + c*x**2)**(5/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(3/2)*(c*x^2 + b*x + a)^(5/2)),x, algorithm="giac")

[Out]

integrate(1/((2*c*d*x + b*d)^(3/2)*(c*x^2 + b*x + a)^(5/2)), x)