3.1347 \(\int \sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=328 \[ \frac{\sqrt{d} \left (b^2-4 a c\right )^{15/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{156 c^4 \sqrt{a+b x+c x^2}}-\frac{\sqrt{d} \left (b^2-4 a c\right )^{15/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{156 c^4 \sqrt{a+b x+c x^2}}+\frac{\left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{156 c^3 d}-\frac{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}}{234 c^2 d}+\frac{\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{3/2}}{13 c d} \]

[Out]

((b^2 - 4*a*c)^2*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(156*c^3*d) - (5*(
b^2 - 4*a*c)*(b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(3/2))/(234*c^2*d) + ((b*d
+ 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(5/2))/(13*c*d) - ((b^2 - 4*a*c)^(15/4)*Sqrt[
d]*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])/(156*c^4*Sqrt[a + b*x + c*x^2]) + ((b^
2 - 4*a*c)^(15/4)*Sqrt[d]*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])/(156*c^4*Sqrt[a
+ b*x + c*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.969658, antiderivative size = 328, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\sqrt{d} \left (b^2-4 a c\right )^{15/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{156 c^4 \sqrt{a+b x+c x^2}}-\frac{\sqrt{d} \left (b^2-4 a c\right )^{15/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{156 c^4 \sqrt{a+b x+c x^2}}+\frac{\left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{156 c^3 d}-\frac{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}}{234 c^2 d}+\frac{\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{3/2}}{13 c d} \]

Antiderivative was successfully verified.

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

[Out]

((b^2 - 4*a*c)^2*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(156*c^3*d) - (5*(
b^2 - 4*a*c)*(b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(3/2))/(234*c^2*d) + ((b*d
+ 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(5/2))/(13*c*d) - ((b^2 - 4*a*c)^(15/4)*Sqrt[
d]*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])/(156*c^4*Sqrt[a + b*x + c*x^2]) + ((b^
2 - 4*a*c)^(15/4)*Sqrt[d]*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])/(156*c^4*Sqrt[a
+ b*x + c*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 166.501, size = 309, normalized size = 0.94 \[ \frac{\left (b d + 2 c d x\right )^{\frac{3}{2}} \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{13 c d} - \frac{5 \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{3}{2}} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{234 c^{2} d} + \frac{\left (- 4 a c + b^{2}\right )^{2} \left (b d + 2 c d x\right )^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}}{156 c^{3} d} - \frac{\sqrt{d} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{15}{4}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{156 c^{4} \sqrt{a + b x + c x^{2}}} + \frac{\sqrt{d} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{15}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{156 c^{4} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*d + 2*c*d*x)**(3/2)*(a + b*x + c*x**2)**(5/2)/(13*c*d) - 5*(-4*a*c + b**2)*(b
*d + 2*c*d*x)**(3/2)*(a + b*x + c*x**2)**(3/2)/(234*c**2*d) + (-4*a*c + b**2)**2
*(b*d + 2*c*d*x)**(3/2)*sqrt(a + b*x + c*x**2)/(156*c**3*d) - sqrt(d)*sqrt(c*(a
+ b*x + c*x**2)/(4*a*c - b**2))*(-4*a*c + b**2)**(15/4)*elliptic_e(asin(sqrt(b*d
+ 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(156*c**4*sqrt(a + b*x + c*x*
*2)) + sqrt(d)*sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*(-4*a*c + b**2)**(15/4)
*elliptic_f(asin(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(156
*c**4*sqrt(a + b*x + c*x**2))

_______________________________________________________________________________________

Mathematica [C]  time = 1.41042, size = 255, normalized size = 0.78 \[ \frac{\sqrt{d (b+2 c x)} \left (c (b+2 c x) (a+x (b+c x)) \left (4 c^2 \left (31 a^2+28 a c x^2+9 c^2 x^4\right )+b^2 \left (26 c^2 x^2-34 a c\right )+8 b c^2 x \left (14 a+9 c x^2\right )+3 b^4-10 b^3 c x\right )+\frac{3 i \left (b^2-4 a c\right )^{7/2} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}}\right )}{468 c^4 \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d*(b + 2*c*x)]*(c*(b + 2*c*x)*(a + x*(b + c*x))*(3*b^4 - 10*b^3*c*x + 8*b*
c^2*x*(14*a + 9*c*x^2) + b^2*(-34*a*c + 26*c^2*x^2) + 4*c^2*(31*a^2 + 28*a*c*x^2
+ 9*c^2*x^4)) + ((3*I)*(b^2 - 4*a*c)^(7/2)*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] - Ellip
ticF[I*ArcSinh[Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]], -1]))/Sqrt[-((b + 2*c*x)
/Sqrt[b^2 - 4*a*c])]))/(468*c^4*Sqrt[a + x*(b + c*x)])

_______________________________________________________________________________________

Maple [B]  time = 0.03, size = 924, normalized size = 2.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/936*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*(1184*x^6*a*c^7+1888*x^4*a^2*c^6+9
92*x^2*a^3*c^5+1128*x^5*b^3*c^5+10*x^2*b^6*c^2+268*x^4*b^4*c^4+6*x*b^7*c+1152*x^
7*b*c^7+1720*x^6*b^2*c^6+768*((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^4*c^4+288*x^8*c^8+248*a^3*b^2*c^3-68*a^2*b^4*c^2+6*
a*b^6*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^8-768*((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^3*b^2*c^3+288*((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*b^4*c^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*b^6*c+3552*x^5*a*b*c^6+3496*x^4*a*b^2
*c^5+3776*x^3*a^2*b*c^5+1072*x^3*a*b^3*c^4+2088*x^2*a^2*b^2*c^4-120*x^2*a*b^4*c^
3+992*x*a^3*b*c^4+200*x*a^2*b^3*c^3-64*x*a*b^5*c^2)/c^4/(2*c^2*x^3+3*b*c*x^2+2*a
*c*x+b^2*x+a*b)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*sqrt(2*c*d*x
+ b*d)*sqrt(c*x^2 + b*x + a), x)

_______________________________________________________________________________________

Sympy [A]  time = 124.336, size = 539, normalized size = 1.64 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*(b*d + 2*c*d*x)**(3/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), (b*d + 2*c*d*x
)**2*exp_polar(I*pi)/(4*c*d**2*polar_lift(a - b**2/(4*c))))*sqrt(polar_lift(a -
b**2/(4*c)))/(4*c*d*gamma(7/4)) - a*b**2*(b*d + 2*c*d*x)**(3/2)*gamma(3/4)*hyper
((-1/2, 3/4), (7/4,), (b*d + 2*c*d*x)**2*exp_polar(I*pi)/(4*c*d**2*polar_lift(a
- b**2/(4*c))))*sqrt(polar_lift(a - b**2/(4*c)))/(8*c**2*d*gamma(7/4)) + a*(b*d
+ 2*c*d*x)**(7/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), (b*d + 2*c*d*x)**2*exp_
polar(I*pi)/(4*c*d**2*polar_lift(a - b**2/(4*c))))*sqrt(polar_lift(a - b**2/(4*c
)))/(8*c**2*d**3*gamma(11/4)) + b**4*(b*d + 2*c*d*x)**(3/2)*gamma(3/4)*hyper((-1
/2, 3/4), (7/4,), (b*d + 2*c*d*x)**2*exp_polar(I*pi)/(4*c*d**2*polar_lift(a - b*
*2/(4*c))))*sqrt(polar_lift(a - b**2/(4*c)))/(64*c**3*d*gamma(7/4)) - b**2*(b*d
+ 2*c*d*x)**(7/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), (b*d + 2*c*d*x)**2*exp_
polar(I*pi)/(4*c*d**2*polar_lift(a - b**2/(4*c))))*sqrt(polar_lift(a - b**2/(4*c
)))/(32*c**3*d**3*gamma(11/4)) + (b*d + 2*c*d*x)**(11/2)*gamma(11/4)*hyper((-1/2
, 11/4), (15/4,), (b*d + 2*c*d*x)**2*exp_polar(I*pi)/(4*c*d**2*polar_lift(a - b*
*2/(4*c))))*sqrt(polar_lift(a - b**2/(4*c)))/(64*c**3*d**5*gamma(15/4))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 1.2836, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done