3.1376 \(\int \frac{1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=274 \[ -\frac{12 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{d^{3/2} \left (b^2-4 a c\right )^{5/4} \sqrt{a+b x+c x^2}}+\frac{12 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{d^{3/2} \left (b^2-4 a c\right )^{5/4} \sqrt{a+b x+c x^2}}-\frac{24 c \sqrt{a+b x+c x^2}}{d \left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x}}-\frac{2}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}} \]

[Out]

-2/((b^2 - 4*a*c)*d*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2]) - (24*c*Sqrt[a +
b*x + c*x^2])/((b^2 - 4*a*c)^2*d*Sqrt[b*d + 2*c*d*x]) + (12*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)^(5/4)*d^(3/2)*Sqrt[a + b*x + c*x^2]) - (12*Sqr
t[-((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)^(5/4)*d^(3/2)*Sqrt[a + b*x +
c*x^2])

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Rubi [A]  time = 0.812328, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{12 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{d^{3/2} \left (b^2-4 a c\right )^{5/4} \sqrt{a+b x+c x^2}}+\frac{12 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{d^{3/2} \left (b^2-4 a c\right )^{5/4} \sqrt{a+b x+c x^2}}-\frac{24 c \sqrt{a+b x+c x^2}}{d \left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x}}-\frac{2}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}} \]

Antiderivative was successfully verified.

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

[Out]

-2/((b^2 - 4*a*c)*d*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2]) - (24*c*Sqrt[a +
b*x + c*x^2])/((b^2 - 4*a*c)^2*d*Sqrt[b*d + 2*c*d*x]) + (12*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)^(5/4)*d^(3/2)*Sqrt[a + b*x + c*x^2]) - (12*Sqr
t[-((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)^(5/4)*d^(3/2)*Sqrt[a + b*x +
c*x^2])

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Rubi in Sympy [A]  time = 151.021, size = 264, normalized size = 0.96 \[ - \frac{24 c \sqrt{a + b x + c x^{2}}}{d \left (- 4 a c + b^{2}\right )^{2} \sqrt{b d + 2 c d x}} - \frac{2}{d \left (- 4 a c + b^{2}\right ) \sqrt{b d + 2 c d x} \sqrt{a + b x + c x^{2}}} + \frac{12 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{d^{\frac{3}{2}} \left (- 4 a c + b^{2}\right )^{\frac{5}{4}} \sqrt{a + b x + c x^{2}}} - \frac{12 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{d^{\frac{3}{2}} \left (- 4 a c + b^{2}\right )^{\frac{5}{4}} \sqrt{a + b x + c x^{2}}} \]

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)**(3/2),x)

[Out]

-24*c*sqrt(a + b*x + c*x**2)/(d*(-4*a*c + b**2)**2*sqrt(b*d + 2*c*d*x)) - 2/(d*(
-4*a*c + b**2)*sqrt(b*d + 2*c*d*x)*sqrt(a + b*x + c*x**2)) + 12*sqrt(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)**(5/4)*sqrt(a + b*x + c*x**2)) -
12*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)**(5/4)*sqrt(a
+ b*x + c*x**2))

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Mathematica [C]  time = 2.32749, size = 253, normalized size = 0.92 \[ \frac{2 i \left (i \left (4 c \left (2 a+3 c x^2\right )+b^2+12 b c x\right )-6 \left (b^2-4 a c\right ) \sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )+6 \left (b^2-4 a c\right ) \sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{d \left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)}} \]

Antiderivative was successfully verified.

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

[Out]

((2*I)*(I*(b^2 + 12*b*c*x + 4*c*(2*a + 3*c*x^2)) + 6*(b^2 - 4*a*c)*Sqrt[-((b + 2
*c*x)/Sqrt[b^2 - 4*a*c])]*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] - 6*(b^2 - 4*a*c)*Sqrt[-((
b + 2*c*x)/Sqrt[b^2 - 4*a*c])]*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Ellipt
icF[I*ArcSinh[Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]], -1]))/((b^2 - 4*a*c)^2*d*
Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)])

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Maple [A]  time = 0.036, size = 339, normalized size = 1.2 \[ 2\,{\frac{\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a}}{{d}^{2} \left ( 2\,{x}^{3}{c}^{2}+3\,{x}^{2}bc+2\,acx+{b}^{2}x+ab \right ) \left ( 4\,ac-{b}^{2} \right ) ^{2}} \left ( 12\,{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) ac\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}-3\,{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){b}^{2}\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}-12\,{c}^{2}{x}^{2}-12\,bxc-8\,ac-{b}^{2} \right ) } \]

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)^(3/2),x)

[Out]

2*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*(12*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*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)-3*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))*b^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)-12*c^2*x^2-12*b*x*c-8*a*c-b^
2)/d^2/(2*c^2*x^3+3*b*c*x^2+2*a*c*x+b^2*x+a*b)/(4*a*c-b^2)^2

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

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)^(3/2)),x, algorithm="maxima")

[Out]

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

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

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)^(3/2)),x, algorithm="fricas")

[Out]

integral(1/((2*c^2*d*x^3 + 3*b*c*d*x^2 + a*b*d + (b^2 + 2*a*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. \[ \int \frac{1}{\left (d \left (b + 2 c x\right )\right )^{\frac{3}{2}} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \]

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)**(3/2),x)

[Out]

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

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

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)^(3/2)),x, algorithm="giac")

[Out]

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