3.1325 \(\int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^{7/2}} \, dx\)

Optimal. Leaf size=283 \[ \frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 c^2 d^{7/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}-\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 c^2 d^{7/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{a+b x+c x^2}}{5 c d^3 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}}-\frac{\sqrt{a+b x+c x^2}}{5 c d (b d+2 c d x)^{5/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.829358, antiderivative size = 283, 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 \[ \frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 c^2 d^{7/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}-\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 c^2 d^{7/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{a+b x+c x^2}}{5 c d^3 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}}-\frac{\sqrt{a+b x+c x^2}}{5 c d (b d+2 c d x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 148.15, size = 265, normalized size = 0.94 \[ - \frac{\sqrt{a + b x + c x^{2}}}{5 c d \left (b d + 2 c d x\right )^{\frac{5}{2}}} + \frac{2 \sqrt{a + b x + c x^{2}}}{5 c d^{3} \left (- 4 a c + b^{2}\right ) \sqrt{b d + 2 c d x}} - \frac{\sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{5 c^{2} d^{\frac{7}{2}} \sqrt [4]{- 4 a c + b^{2}} \sqrt{a + b x + c x^{2}}} + \frac{\sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{5 c^{2} d^{\frac{7}{2}} \sqrt [4]{- 4 a c + b^{2}} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(1/2)/(2*c*d*x+b*d)**(7/2),x)

[Out]

-sqrt(a + b*x + c*x**2)/(5*c*d*(b*d + 2*c*d*x)**(5/2)) + 2*sqrt(a + b*x + c*x**2
)/(5*c*d**3*(-4*a*c + b**2)*sqrt(b*d + 2*c*d*x)) - 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)/(5*c**2*d**(7/2)*(-4*a*c + b**2)**(1/4)*sqrt(a + b*x + c*x**2)) + 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)/(5*c**2*d**(7/2)*(-4*a*c + b**2)**(1/4)*sqrt(a +
b*x + c*x**2))

_______________________________________________________________________________________

Mathematica [C]  time = 1.41348, size = 208, normalized size = 0.73 \[ \frac{-\frac{c (a+x (b+c x)) \left (4 c \left (a+2 c x^2\right )+b^2+8 b c x\right )}{4 a c-b^2}-i (b+2 c x)^2 \sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{5 c^2 d \sqrt{a+x (b+c x)} (d (b+2 c x))^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-((c*(a + x*(b + c*x))*(b^2 + 8*b*c*x + 4*c*(a + 2*c*x^2)))/(-b^2 + 4*a*c)) - I
*(b + 2*c*x)^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] - EllipticF[I*ArcSinh[Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]], -1]))/(5*c^2*d
*(d*(b + 2*c*x))^(5/2)*Sqrt[a + x*(b + c*x)])

_______________________________________________________________________________________

Maple [B]  time = 0.067, size = 874, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^(7/2),x)

[Out]

1/10*(16*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)-4*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)+16*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)-4*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)+4*((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-((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^4-16*c^4*x^4-32*b*c^3*x^
3-24*x^2*a*c^3-18*x^2*b^2*c^2-24*x*a*b*c^2-2*b^3*c*x-8*a^2*c^2-2*a*c*b^2)*(d*(2*
c*x+b))^(1/2)/d^4/(c*x^2+b*x+a)^(1/2)/(4*a*c-b^2)/(2*c*x+b)^3/c^2

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^(7/2),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^(7/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^2 + b*x + a)/((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. \[ \int \frac{\sqrt{a + b x + c x^{2}}}{\left (d \left (b + 2 c x\right )\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**(1/2)/(2*c*d*x+b*d)**(7/2),x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^(7/2),x, algorithm="giac")

[Out]

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