3.1343 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{13/2}} \, dx\)

Optimal. Leaf size=211 \[ \frac{5 \sqrt [4]{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{308 c^4 d^{13/2} \sqrt{a+b x+c x^2}}-\frac{5 \sqrt{a+b x+c x^2}}{308 c^3 d^5 (b d+2 c d x)^{3/2}}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{154 c^2 d^3 (b d+2 c d x)^{7/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{11 c d (b d+2 c d x)^{11/2}} \]

[Out]

(-5*Sqrt[a + b*x + c*x^2])/(308*c^3*d^5*(b*d + 2*c*d*x)^(3/2)) - (5*(a + b*x + c
*x^2)^(3/2))/(154*c^2*d^3*(b*d + 2*c*d*x)^(7/2)) - (a + b*x + c*x^2)^(5/2)/(11*c
*d*(b*d + 2*c*d*x)^(11/2)) + (5*(b^2 - 4*a*c)^(1/4)*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])/(308*c^4*d^(13/2)*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.509696, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{5 \sqrt [4]{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{308 c^4 d^{13/2} \sqrt{a+b x+c x^2}}-\frac{5 \sqrt{a+b x+c x^2}}{308 c^3 d^5 (b d+2 c d x)^{3/2}}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{154 c^2 d^3 (b d+2 c d x)^{7/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{11 c d (b d+2 c d x)^{11/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(13/2),x]

[Out]

(-5*Sqrt[a + b*x + c*x^2])/(308*c^3*d^5*(b*d + 2*c*d*x)^(3/2)) - (5*(a + b*x + c
*x^2)^(3/2))/(154*c^2*d^3*(b*d + 2*c*d*x)^(7/2)) - (a + b*x + c*x^2)^(5/2)/(11*c
*d*(b*d + 2*c*d*x)^(11/2)) + (5*(b^2 - 4*a*c)^(1/4)*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])/(308*c^4*d^(13/2)*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 120.101, size = 202, normalized size = 0.96 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{11 c d \left (b d + 2 c d x\right )^{\frac{11}{2}}} - \frac{5 \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{154 c^{2} d^{3} \left (b d + 2 c d x\right )^{\frac{7}{2}}} - \frac{5 \sqrt{a + b x + c x^{2}}}{308 c^{3} d^{5} \left (b d + 2 c d x\right )^{\frac{3}{2}}} + \frac{5 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \sqrt [4]{- 4 a c + b^{2}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{308 c^{4} d^{\frac{13}{2}} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**(13/2),x)

[Out]

-(a + b*x + c*x**2)**(5/2)/(11*c*d*(b*d + 2*c*d*x)**(11/2)) - 5*(a + b*x + c*x**
2)**(3/2)/(154*c**2*d**3*(b*d + 2*c*d*x)**(7/2)) - 5*sqrt(a + b*x + c*x**2)/(308
*c**3*d**5*(b*d + 2*c*d*x)**(3/2)) + 5*sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))
*(-4*a*c + b**2)**(1/4)*elliptic_f(asin(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b
**2)**(1/4))), -1)/(308*c**4*d**(13/2)*sqrt(a + b*x + c*x**2))

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Mathematica [C]  time = 0.931821, size = 187, normalized size = 0.89 \[ \frac{-\frac{(b+2 c x) (a+x (b+c x)) \left (-24 \left (b^2-4 a c\right ) (b+2 c x)^2+7 \left (b^2-4 a c\right )^2+37 (b+2 c x)^4\right )}{4 c^3}+\frac{5 i (b+2 c x)^{15/2} \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{c^4 \sqrt{-\sqrt{b^2-4 a c}}}}{308 \sqrt{a+x (b+c x)} (d (b+2 c x))^{13/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(13/2),x]

[Out]

(-((b + 2*c*x)*(a + x*(b + c*x))*(7*(b^2 - 4*a*c)^2 - 24*(b^2 - 4*a*c)*(b + 2*c*
x)^2 + 37*(b + 2*c*x)^4))/(4*c^3) + ((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])/(c^4*Sqrt[-Sqrt[b^2 - 4*a*c]]))/(308*(d*(b + 2*c*x))^(13/2)*Sqrt[a +
x*(b + c*x)])

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Maple [B]  time = 0.034, size = 1035, normalized size = 4.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^(13/2),x)

[Out]

1/616*(c*x^2+b*x+a)^(1/2)*(d*(2*c*x+b))^(1/2)*(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^2-296*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-888*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^5-488*x^4*a*c^5-988*x^4*b^2*c^4-976*x^3*a*b*c^4-496*b^3*c^3*x^3-248*x^2*
a^2*c^4-608*x^2*a*b^2*c^3-110*x^2*b^4*c^2-248*a^2*b*c^3*x-120*a*b^3*c^2*x-10*b^5
*c*x-56*a^3*c^3-20*a^2*b^2*c^2-10*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/c^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(13/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(13/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(13/2),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(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)**(5/2)/(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. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(13/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(13/2), x)