3.1331 \(\int \frac{\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{13/2}} \, dx\)

Optimal. Leaf size=221 \[ \frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{154 c^3 d^{13/2} \left (b^2-4 a c\right )^{3/4} \sqrt{a+b x+c x^2}}+\frac{\sqrt{a+b x+c x^2}}{77 c^2 d^5 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}-\frac{3 \sqrt{a+b x+c x^2}}{154 c^2 d^3 (b d+2 c d x)^{7/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{11 c d (b d+2 c d x)^{11/2}} \]

[Out]

(-3*Sqrt[a + b*x + c*x^2])/(154*c^2*d^3*(b*d + 2*c*d*x)^(7/2)) + Sqrt[a + b*x +
c*x^2]/(77*c^2*(b^2 - 4*a*c)*d^5*(b*d + 2*c*d*x)^(3/2)) - (a + b*x + c*x^2)^(3/2
)/(11*c*d*(b*d + 2*c*d*x)^(11/2)) + (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])/(154
*c^3*(b^2 - 4*a*c)^(3/4)*d^(13/2)*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.518363, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{154 c^3 d^{13/2} \left (b^2-4 a c\right )^{3/4} \sqrt{a+b x+c x^2}}+\frac{\sqrt{a+b x+c x^2}}{77 c^2 d^5 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}-\frac{3 \sqrt{a+b x+c x^2}}{154 c^2 d^3 (b d+2 c d x)^{7/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{11 c d (b d+2 c d x)^{11/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^(13/2),x]

[Out]

(-3*Sqrt[a + b*x + c*x^2])/(154*c^2*d^3*(b*d + 2*c*d*x)^(7/2)) + Sqrt[a + b*x +
c*x^2]/(77*c^2*(b^2 - 4*a*c)*d^5*(b*d + 2*c*d*x)^(3/2)) - (a + b*x + c*x^2)^(3/2
)/(11*c*d*(b*d + 2*c*d*x)^(11/2)) + (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])/(154
*c^3*(b^2 - 4*a*c)^(3/4)*d^(13/2)*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 121.502, size = 207, normalized size = 0.94 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{11 c d \left (b d + 2 c d x\right )^{\frac{11}{2}}} - \frac{3 \sqrt{a + b x + c x^{2}}}{154 c^{2} d^{3} \left (b d + 2 c d x\right )^{\frac{7}{2}}} + \frac{\sqrt{a + b x + c x^{2}}}{77 c^{2} d^{5} \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{3}{2}}} + \frac{\sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{154 c^{3} d^{\frac{13}{2}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(3/2)/(2*c*d*x+b*d)**(13/2),x)

[Out]

-(a + b*x + c*x**2)**(3/2)/(11*c*d*(b*d + 2*c*d*x)**(11/2)) - 3*sqrt(a + b*x + c
*x**2)/(154*c**2*d**3*(b*d + 2*c*d*x)**(7/2)) + sqrt(a + b*x + c*x**2)/(77*c**2*
d**5*(-4*a*c + b**2)*(b*d + 2*c*d*x)**(3/2)) + 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)/(154*c**3*d**(13/2)*(-4*a*c + b**2)**(3/4)*sqrt(a + b*x + c*x**2))

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Mathematica [C]  time = 0.972702, size = 192, normalized size = 0.87 \[ \frac{c (b+2 c x) (a+x (b+c x)) \left (-13 \left (b^2-4 a c\right ) (b+2 c x)^2+7 \left (b^2-4 a c\right )^2+4 (b+2 c x)^4\right )+\frac{2 i (b+2 c x)^{15/2} \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{\sqrt{-\sqrt{b^2-4 a c}}}}{308 c^3 \left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} (d (b+2 c x))^{13/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^(13/2),x]

[Out]

(c*(b + 2*c*x)*(a + x*(b + c*x))*(7*(b^2 - 4*a*c)^2 - 13*(b^2 - 4*a*c)*(b + 2*c*
x)^2 + 4*(b + 2*c*x)^4) + ((2*I)*(b + 2*c*x)^(15/2)*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]
)/Sqrt[-Sqrt[b^2 - 4*a*c]])/(308*c^3*(b^2 - 4*a*c)*(d*(b + 2*c*x))^(13/2)*Sqrt[a
+ x*(b + c*x)])

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Maple [B]  time = 0.039, size = 1046, normalized size = 4.7 \[ \text{result too large to display} \]

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)^(13/2),x)

[Out]

-1/308*(c*x^2+b*x+a)^(1/2)*(d*(2*c*x+b))^(1/2)*(32*((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^5*c^5+80*(
(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^4*b*c^4+80*((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^3*b^2*c^3+40*((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^
2*b^3*c^2+64*c^6*x^6+10*((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*b^4*c+192*b*c^5*x^5+((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^5+272*x^4*a*c^5+172*x^4*b^2*c^4+544*x^3*a*b*c^4+24*b^3*c^3*x^3+320*x^2*a^2*c^4
+248*x^2*a*b^2*c^3-22*x^2*b^4*c^2+320*a^2*b*c^3*x-24*a*b^3*c^2*x-2*b^5*c*x+112*a
^3*c^3-4*a^2*b^2*c^2-2*a*b^4*c)/d^7/(2*c^2*x^3+3*b*c*x^2+2*a*c*x+b^2*x+a*b)/(2*c
*x+b)^5/(4*a*c-b^2)/c^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{13}{2}}}\,{d x} \]

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)^(13/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^(13/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{{\left (64 \, c^{6} d^{6} x^{6} + 192 \, b c^{5} d^{6} x^{5} + 240 \, b^{2} c^{4} d^{6} x^{4} + 160 \, b^{3} c^{3} d^{6} x^{3} + 60 \, b^{4} c^{2} d^{6} x^{2} + 12 \, b^{5} c d^{6} x + b^{6} d^{6}\right )} \sqrt{2 \, c d x + b d}}, x\right ) \]

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)^(13/2),x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x + a)^(3/2)/((64*c^6*d^6*x^6 + 192*b*c^5*d^6*x^5 + 240*b^2*
c^4*d^6*x^4 + 160*b^3*c^3*d^6*x^3 + 60*b^4*c^2*d^6*x^2 + 12*b^5*c*d^6*x + b^6*d^
6)*sqrt(2*c*d*x + b*d)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**(13/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{13}{2}}}\,{d x} \]

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)^(13/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^(13/2), x)