3.1355 \(\int \frac{1}{(b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=144 \[ \frac{2 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{3 c d^{5/2} \left (b^2-4 a c\right )^{3/4} \sqrt{a+b x+c x^2}}+\frac{4 \sqrt{a+b x+c x^2}}{3 d \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.344674, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{2 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{3 c d^{5/2} \left (b^2-4 a c\right )^{3/4} \sqrt{a+b x+c x^2}}+\frac{4 \sqrt{a+b x+c x^2}}{3 d \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 77.9743, size = 134, normalized size = 0.93 \[ \frac{4 \sqrt{a + b x + c x^{2}}}{3 d \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{3}{2}}} + \frac{2 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{3 c d^{\frac{5}{2}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.526347, size = 167, normalized size = 1.16 \[ \frac{4 c \sqrt{-\sqrt{b^2-4 a c}} (a+x (b+c x))+2 i (b+2 c x)^{5/2} \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{3 c d \sqrt{-\sqrt{b^2-4 a c}} \left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} (d (b+2 c x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.031, size = 362, normalized size = 2.5 \[ -{\frac{1}{3\,{d}^{3} \left ( 2\,{x}^{3}{c}^{2}+3\,{x}^{2}bc+2\,acx+{b}^{2}x+ab \right ) \left ( 4\,ac-{b}^{2} \right ) c \left ( 2\,cx+b \right ) }\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a} \left ( 2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}xc+\sqrt{{1 \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{-{(2\,cx+b){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{{1 \left ( -b-2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}{\it EllipticF} \left ({\frac{\sqrt{2}}{2}\sqrt{{1 \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}b+4\,{c}^{2}{x}^{2}+4\,bxc+4\,ac \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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