3.1321 \(\int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^{13/2}} \, dx\)

Optimal. Leaf size=231 \[ \frac{5 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{231 c^2 d^{13/2} \left (b^2-4 a c\right )^{7/4} \sqrt{a+b x+c x^2}}+\frac{10 \sqrt{a+b x+c x^2}}{231 c d^5 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}+\frac{2 \sqrt{a+b x+c x^2}}{77 c d^3 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}-\frac{\sqrt{a+b x+c x^2}}{11 c d (b d+2 c d x)^{11/2}} \]

[Out]

-Sqrt[a + b*x + c*x^2]/(11*c*d*(b*d + 2*c*d*x)^(11/2)) + (2*Sqrt[a + b*x + c*x^2
])/(77*c*(b^2 - 4*a*c)*d^3*(b*d + 2*c*d*x)^(7/2)) + (10*Sqrt[a + b*x + c*x^2])/(
231*c*(b^2 - 4*a*c)^2*d^5*(b*d + 2*c*d*x)^(3/2)) + (5*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)*Sqr
t[d])], -1])/(231*c^2*(b^2 - 4*a*c)^(7/4)*d^(13/2)*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.545238, antiderivative size = 231, 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 \[ \frac{5 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{231 c^2 d^{13/2} \left (b^2-4 a c\right )^{7/4} \sqrt{a+b x+c x^2}}+\frac{10 \sqrt{a+b x+c x^2}}{231 c d^5 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}+\frac{2 \sqrt{a+b x+c x^2}}{77 c d^3 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}-\frac{\sqrt{a+b x+c x^2}}{11 c d (b d+2 c d x)^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[a + b*x + c*x^2]/(11*c*d*(b*d + 2*c*d*x)^(11/2)) + (2*Sqrt[a + b*x + c*x^2
])/(77*c*(b^2 - 4*a*c)*d^3*(b*d + 2*c*d*x)^(7/2)) + (10*Sqrt[a + b*x + c*x^2])/(
231*c*(b^2 - 4*a*c)^2*d^5*(b*d + 2*c*d*x)^(3/2)) + (5*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)*Sqr
t[d])], -1])/(231*c^2*(b^2 - 4*a*c)^(7/4)*d^(13/2)*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 116.28, size = 218, normalized size = 0.94 \[ - \frac{\sqrt{a + b x + c x^{2}}}{11 c d \left (b d + 2 c d x\right )^{\frac{11}{2}}} + \frac{2 \sqrt{a + b x + c x^{2}}}{77 c d^{3} \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{7}{2}}} + \frac{10 \sqrt{a + b x + c x^{2}}}{231 c d^{5} \left (- 4 a c + b^{2}\right )^{2} \left (b d + 2 c d x\right )^{\frac{3}{2}}} + \frac{5 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{231 c^{2} d^{\frac{13}{2}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(1/2)/(2*c*d*x+b*d)**(13/2),x)

[Out]

-sqrt(a + b*x + c*x**2)/(11*c*d*(b*d + 2*c*d*x)**(11/2)) + 2*sqrt(a + b*x + c*x*
*2)/(77*c*d**3*(-4*a*c + b**2)*(b*d + 2*c*d*x)**(7/2)) + 10*sqrt(a + b*x + c*x**
2)/(231*c*d**5*(-4*a*c + b**2)**2*(b*d + 2*c*d*x)**(3/2)) + 5*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)/(231*c**2*d**(13/2)*(-4*a*c + b**2)**(7/4)*sqrt(a + b*x + c*x
**2))

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Mathematica [C]  time = 0.893898, size = 193, normalized size = 0.84 \[ \frac{-c (b+2 c x) (a+x (b+c x)) \left (-6 \left (b^2-4 a c\right ) (b+2 c x)^2+21 \left (b^2-4 a c\right )^2-10 (b+2 c x)^4\right )+\frac{5 i (b+2 c x)^{15/2} \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{\sqrt{-\sqrt{b^2-4 a c}}}}{231 c^2 \left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)} (d (b+2 c x))^{13/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-(c*(b + 2*c*x)*(a + x*(b + c*x))*(21*(b^2 - 4*a*c)^2 - 6*(b^2 - 4*a*c)*(b + 2*
c*x)^2 - 10*(b + 2*c*x)^4)) + ((5*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]])/(231*c^2*(b^2 - 4*a*c)^2*(d*(b + 2*c*x))^(13/2)*
Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.071, size = 1016, normalized size = 4.4 \[ \text{result too large to display} \]

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)^(13/2),x)

[Out]

1/462*(160*((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+400*((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+400*((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+200*((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+320*c^6*x^6+50*((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+960*b*c^5*x^5+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+128*x^4*a*c^5+1168*x^4*b^2*c^4
+256*x^3*a*b*c^4+736*b^3*c^3*x^3-864*x^2*a^2*c^4+624*x^2*a*b^2*c^3+198*x^2*b^4*c
^2-864*a^2*b*c^3*x+496*a*b^3*c^2*x-10*b^5*c*x-672*a^3*c^3+288*a^2*b^2*c^2-10*a*b
^4*c)/d^7*(d*(2*c*x+b))^(1/2)/(c*x^2+b*x+a)^(1/2)/(4*a*c-b^2)^2/(2*c*x+b)^6/c^2

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

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)^(13/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^(13/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^2 + b*x + a)/((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)**(1/2)/(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. \[ \int \frac{\sqrt{c x^{2} + b x + a}}{{\left (2 \, c d x + b d\right )}^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^(13/2),x, algorithm="giac")

[Out]

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