3.1336 \(\int \frac{\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{7/2}} \, dx\)

Optimal. Leaf size=273 \[ -\frac{3 \left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{10 c^3 d^{7/2} \sqrt{a+b x+c x^2}}+\frac{3 \left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{10 c^3 d^{7/2} \sqrt{a+b x+c x^2}}-\frac{3 \sqrt{a+b x+c x^2}}{10 c^2 d^3 \sqrt{b d+2 c d x}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{5 c d (b d+2 c d x)^{5/2}} \]

[Out]

(-3*Sqrt[a + b*x + c*x^2])/(10*c^2*d^3*Sqrt[b*d + 2*c*d*x]) - (a + b*x + c*x^2)^
(3/2)/(5*c*d*(b*d + 2*c*d*x)^(5/2)) + (3*(b^2 - 4*a*c)^(3/4)*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^(7/2)*Sqrt[a + b*x + c*x^2]) - (3*(b^2 - 4*a*c)^(3
/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^(7/2)*Sqrt[a + b*x + c*x^2]
)

_______________________________________________________________________________________

Rubi [A]  time = 0.817612, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{3 \left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{10 c^3 d^{7/2} \sqrt{a+b x+c x^2}}+\frac{3 \left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{10 c^3 d^{7/2} \sqrt{a+b x+c x^2}}-\frac{3 \sqrt{a+b x+c x^2}}{10 c^2 d^3 \sqrt{b d+2 c d x}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{5 c d (b d+2 c d x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^(7/2),x]

[Out]

(-3*Sqrt[a + b*x + c*x^2])/(10*c^2*d^3*Sqrt[b*d + 2*c*d*x]) - (a + b*x + c*x^2)^
(3/2)/(5*c*d*(b*d + 2*c*d*x)^(5/2)) + (3*(b^2 - 4*a*c)^(3/4)*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^(7/2)*Sqrt[a + b*x + c*x^2]) - (3*(b^2 - 4*a*c)^(3
/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^(7/2)*Sqrt[a + b*x + c*x^2]
)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 145.608, size = 262, normalized size = 0.96 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{5 c d \left (b d + 2 c d x\right )^{\frac{5}{2}}} - \frac{3 \sqrt{a + b x + c x^{2}}}{10 c^{2} d^{3} \sqrt{b d + 2 c d x}} + \frac{3 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{10 c^{3} d^{\frac{7}{2}} \sqrt{a + b x + c x^{2}}} - \frac{3 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{10 c^{3} d^{\frac{7}{2}} \sqrt{a + b x + c x^{2}}} \]

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)**(7/2),x)

[Out]

-(a + b*x + c*x**2)**(3/2)/(5*c*d*(b*d + 2*c*d*x)**(5/2)) - 3*sqrt(a + b*x + c*x
**2)/(10*c**2*d**3*sqrt(b*d + 2*c*d*x)) + 3*sqrt(c*(a + b*x + c*x**2)/(4*a*c - b
**2))*(-4*a*c + b**2)**(3/4)*elliptic_e(asin(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*
c + b**2)**(1/4))), -1)/(10*c**3*d**(7/2)*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)**(3/4)*elliptic_f(asin(sqrt(b*d
+ 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(10*c**3*d**(7/2)*sqrt(a + b*x
+ c*x**2))

_______________________________________________________________________________________

Mathematica [C]  time = 1.64742, size = 198, normalized size = 0.73 \[ \frac{-c (a+x (b+c x)) \left (2 c \left (a+7 c x^2\right )+3 b^2+14 b c x\right )+\frac{3 i (b+2 c x)^4 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{\left (-\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )^{3/2}}}{10 c^3 d \sqrt{a+x (b+c x)} (d (b+2 c x))^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^(7/2),x]

[Out]

(-(c*(a + x*(b + c*x))*(3*b^2 + 14*b*c*x + 2*c*(a + 7*c*x^2))) + ((3*I)*(b + 2*c
*x)^4*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)/Sqrt
[b^2 - 4*a*c])]], -1]))/(-((b + 2*c*x)/Sqrt[b^2 - 4*a*c]))^(3/2))/(10*c^3*d*(d*(
b + 2*c*x))^(5/2)*Sqrt[a + x*(b + c*x)])

_______________________________________________________________________________________

Maple [B]  time = 0.043, size = 893, normalized size = 3.3 \[ \text{result too large to display} \]

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)^(7/2),x)

[Out]

1/20*(c*x^2+b*x+a)^(1/2)*(d*(2*c*x+b))^(1/2)*(48*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)-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))*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)+48*Ellipti
cE(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)
-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))*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)+12*((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^4-28*c^4*x^4-56*b*c^3*x^3-32*x^2*a*c^3-34*x^2*b^2*c^2-32*x*
a*b*c^2-6*b^3*c*x-4*a^2*c^2-6*a*c*b^2)/d^4/(2*c^2*x^3+3*b*c*x^2+2*a*c*x+b^2*x+a*
b)/(2*c*x+b)^2/c^3

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{7}{2}}}\,{d x} \]

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)^(7/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^(7/2), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{{\left (8 \, c^{3} d^{3} x^{3} + 12 \, b c^{2} d^{3} x^{2} + 6 \, b^{2} c d^{3} x + b^{3} d^{3}\right )} \sqrt{2 \, c d x + b d}}, x\right ) \]

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)^(7/2),x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x + a)^(3/2)/((8*c^3*d^3*x^3 + 12*b*c^2*d^3*x^2 + 6*b^2*c*d^
3*x + b^3*d^3)*sqrt(2*c*d*x + b*d)), x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{\left (d \left (b + 2 c x\right )\right )^{\frac{7}{2}}}\, dx \]

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)**(7/2),x)

[Out]

Integral((a + b*x + c*x**2)**(3/2)/(d*(b + 2*c*x))**(7/2), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{7}{2}}}\,{d x} \]

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)^(7/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^(7/2), x)