3.1335 \(\int \frac{\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{3/2}} \, dx\)

Optimal. Leaf size=271 \[ \frac{3 \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{10 c^3 d^{3/2} \sqrt{a+b x+c x^2}}-\frac{3 \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{10 c^3 d^{3/2} \sqrt{a+b x+c x^2}}+\frac{3 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{10 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{3/2}}{c d \sqrt{b d+2 c d x}} \]

[Out]

(3*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(10*c^2*d^3) - (a + b*x + c*x^2)
^(3/2)/(c*d*Sqrt[b*d + 2*c*d*x]) - (3*(b^2 - 4*a*c)^(7/4)*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])/(10*c^3*d^(3/2)*Sqrt[a + b*x + c*x^2]) + (3*(b^2 - 4*a*c)^(7/4)
*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])/(10*c^3*d^(3/2)*Sqrt[a + b*x + c*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.823141, antiderivative size = 271, 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 \[ \frac{3 \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{10 c^3 d^{3/2} \sqrt{a+b x+c x^2}}-\frac{3 \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{10 c^3 d^{3/2} \sqrt{a+b x+c x^2}}+\frac{3 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{10 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{3/2}}{c d \sqrt{b d+2 c d x}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(10*c^2*d^3) - (a + b*x + c*x^2)
^(3/2)/(c*d*Sqrt[b*d + 2*c*d*x]) - (3*(b^2 - 4*a*c)^(7/4)*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])/(10*c^3*d^(3/2)*Sqrt[a + b*x + c*x^2]) + (3*(b^2 - 4*a*c)^(7/4)
*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])/(10*c^3*d^(3/2)*Sqrt[a + b*x + c*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 145.967, size = 260, normalized size = 0.96 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{c d \sqrt{b d + 2 c d x}} + \frac{3 \left (b d + 2 c d x\right )^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}}{10 c^{2} d^{3}} - \frac{3 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{10 c^{3} d^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}} + \frac{3 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{10 c^{3} d^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x + c*x**2)**(3/2)/(c*d*sqrt(b*d + 2*c*d*x)) + 3*(b*d + 2*c*d*x)**(3/2)*
sqrt(a + b*x + c*x**2)/(10*c**2*d**3) - 3*sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**
2))*(-4*a*c + b**2)**(7/4)*elliptic_e(asin(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c
+ b**2)**(1/4))), -1)/(10*c**3*d**(3/2)*sqrt(a + b*x + c*x**2)) + 3*sqrt(c*(a +
b*x + c*x**2)/(4*a*c - b**2))*(-4*a*c + b**2)**(7/4)*elliptic_f(asin(sqrt(b*d +
2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(10*c**3*d**(3/2)*sqrt(a + b*x +
c*x**2))

_______________________________________________________________________________________

Mathematica [C]  time = 1.36797, size = 200, normalized size = 0.74 \[ \frac{c (a+x (b+c x)) \left (2 c \left (c x^2-5 a\right )+3 b^2+2 b c x\right )-3 i \left (b^2-4 a c\right )^2 \sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{10 c^3 d \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(c*(a + x*(b + c*x))*(3*b^2 + 2*b*c*x + 2*c*(-5*a + c*x^2)) - (3*I)*(b^2 - 4*a*c
)^2*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] - Ellipt
icF[I*ArcSinh[Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]], -1]))/(10*c^3*d*Sqrt[d*(b
+ 2*c*x)]*Sqrt[a + x*(b + c*x)])

_______________________________________________________________________________________

Maple [B]  time = 0.042, size = 504, normalized size = 1.9 \[{\frac{1}{20\,{d}^{2} \left ( 2\,{x}^{3}{c}^{2}+3\,{x}^{2}bc+2\,acx+{b}^{2}x+ab \right ){c}^{3}}\sqrt{c{x}^{2}+bx+a}\sqrt{d \left ( 2\,cx+b \right ) } \left ( 48\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){a}^{2}{c}^{2}-24\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) a{b}^{2}c+3\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){b}^{4}+4\,{c}^{4}{x}^{4}+8\,b{c}^{3}{x}^{3}-16\,{x}^{2}a{c}^{3}+10\,{x}^{2}{b}^{2}{c}^{2}-16\,xab{c}^{2}+6\,{b}^{3}cx-20\,{a}^{2}{c}^{2}+6\,ac{b}^{2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*(c*x^2+b*x+a)^(1/2)*(d*(2*c*x+b))^(1/2)*(48*((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^2-24*((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+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)*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^4+4*c^4*x^4+8*
b*c^3*x^3-16*x^2*a*c^3+10*x^2*b^2*c^2-16*x*a*b*c^2+6*b^3*c*x-20*a^2*c^2+6*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)/c^3

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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