3.1345 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{21/2}} \, dx\)

Optimal. Leaf size=305 \[ \frac{5 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{17556 c^4 d^{21/2} \left (b^2-4 a c\right )^{7/4} \sqrt{a+b x+c x^2}}+\frac{5 \sqrt{a+b x+c x^2}}{8778 c^3 d^9 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}+\frac{\sqrt{a+b x+c x^2}}{2926 c^3 d^7 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}-\frac{\sqrt{a+b x+c x^2}}{836 c^3 d^5 (b d+2 c d x)^{11/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{114 c^2 d^3 (b d+2 c d x)^{15/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{19 c d (b d+2 c d x)^{19/2}} \]

[Out]

-Sqrt[a + b*x + c*x^2]/(836*c^3*d^5*(b*d + 2*c*d*x)^(11/2)) + Sqrt[a + b*x + c*x
^2]/(2926*c^3*(b^2 - 4*a*c)*d^7*(b*d + 2*c*d*x)^(7/2)) + (5*Sqrt[a + b*x + c*x^2
])/(8778*c^3*(b^2 - 4*a*c)^2*d^9*(b*d + 2*c*d*x)^(3/2)) - (a + b*x + c*x^2)^(3/2
)/(114*c^2*d^3*(b*d + 2*c*d*x)^(15/2)) - (a + b*x + c*x^2)^(5/2)/(19*c*d*(b*d +
2*c*d*x)^(19/2)) + (5*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[Arc
Sin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(17556*c^4*(b^2 - 4
*a*c)^(7/4)*d^(21/2)*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.757223, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{5 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{17556 c^4 d^{21/2} \left (b^2-4 a c\right )^{7/4} \sqrt{a+b x+c x^2}}+\frac{5 \sqrt{a+b x+c x^2}}{8778 c^3 d^9 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}+\frac{\sqrt{a+b x+c x^2}}{2926 c^3 d^7 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}-\frac{\sqrt{a+b x+c x^2}}{836 c^3 d^5 (b d+2 c d x)^{11/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{114 c^2 d^3 (b d+2 c d x)^{15/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{19 c d (b d+2 c d x)^{19/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(21/2),x]

[Out]

-Sqrt[a + b*x + c*x^2]/(836*c^3*d^5*(b*d + 2*c*d*x)^(11/2)) + Sqrt[a + b*x + c*x
^2]/(2926*c^3*(b^2 - 4*a*c)*d^7*(b*d + 2*c*d*x)^(7/2)) + (5*Sqrt[a + b*x + c*x^2
])/(8778*c^3*(b^2 - 4*a*c)^2*d^9*(b*d + 2*c*d*x)^(3/2)) - (a + b*x + c*x^2)^(3/2
)/(114*c^2*d^3*(b*d + 2*c*d*x)^(15/2)) - (a + b*x + c*x^2)^(5/2)/(19*c*d*(b*d +
2*c*d*x)^(19/2)) + (5*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[Arc
Sin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(17556*c^4*(b^2 - 4
*a*c)^(7/4)*d^(21/2)*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 160.705, size = 291, normalized size = 0.95 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{19 c d \left (b d + 2 c d x\right )^{\frac{19}{2}}} - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{114 c^{2} d^{3} \left (b d + 2 c d x\right )^{\frac{15}{2}}} - \frac{\sqrt{a + b x + c x^{2}}}{836 c^{3} d^{5} \left (b d + 2 c d x\right )^{\frac{11}{2}}} + \frac{\sqrt{a + b x + c x^{2}}}{2926 c^{3} d^{7} \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{7}{2}}} + \frac{5 \sqrt{a + b x + c x^{2}}}{8778 c^{3} d^{9} \left (- 4 a c + b^{2}\right )^{2} \left (b d + 2 c d x\right )^{\frac{3}{2}}} + \frac{5 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{17556 c^{4} d^{\frac{21}{2}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**(21/2),x)

[Out]

-(a + b*x + c*x**2)**(5/2)/(19*c*d*(b*d + 2*c*d*x)**(19/2)) - (a + b*x + c*x**2)
**(3/2)/(114*c**2*d**3*(b*d + 2*c*d*x)**(15/2)) - sqrt(a + b*x + c*x**2)/(836*c*
*3*d**5*(b*d + 2*c*d*x)**(11/2)) + sqrt(a + b*x + c*x**2)/(2926*c**3*d**7*(-4*a*
c + b**2)*(b*d + 2*c*d*x)**(7/2)) + 5*sqrt(a + b*x + c*x**2)/(8778*c**3*d**9*(-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)
/(17556*c**4*d**(21/2)*(-4*a*c + b**2)**(7/4)*sqrt(a + b*x + c*x**2))

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Mathematica [C]  time = 1.58129, size = 233, normalized size = 0.76 \[ \frac{-c (b+2 c x) (a+x (b+c x)) \left (-24 \left (b^2-4 a c\right ) (b+2 c x)^6+469 \left (b^2-4 a c\right )^2 (b+2 c x)^4-616 \left (b^2-4 a c\right )^3 (b+2 c x)^2+231 \left (b^2-4 a c\right )^4-40 (b+2 c x)^8\right )+\frac{20 i (b+2 c x)^{23/2} \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{\sqrt{-\sqrt{b^2-4 a c}}}}{70224 c^4 \left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)} (d (b+2 c x))^{21/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(21/2),x]

[Out]

(-(c*(b + 2*c*x)*(a + x*(b + c*x))*(231*(b^2 - 4*a*c)^4 - 616*(b^2 - 4*a*c)^3*(b
+ 2*c*x)^2 + 469*(b^2 - 4*a*c)^2*(b + 2*c*x)^4 - 24*(b^2 - 4*a*c)*(b + 2*c*x)^6
- 40*(b + 2*c*x)^8)) + ((20*I)*(b + 2*c*x)^(23/2)*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]])/(70224*c^4*(b^2 - 4*a*c)^2*(d*(b + 2*c*x))^(21/2)*Sqr
t[a + x*(b + c*x)])

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Maple [B]  time = 0.087, size = 1843, normalized size = 6. \[ \text{result too large to display} \]

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)^(21/2),x)

[Out]

1/35112*(c*x^2+b*x+a)^(1/2)*(d*(2*c*x+b))^(1/2)*(9856*a^4*b^2*c^4-56*a^3*b^4*c^3
-20*a^2*b^6*c^2-10*a*b^8*c+2560*((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^9*c^9-29568*a^5*c^5+8192*x^7*
a*b*c^8+45888*x^6*a*b^2*c^7-189312*x^5*a^2*b*c^7+108992*x^5*a*b^3*c^6-132480*x^4
*a^2*b^2*c^6+101240*x^4*a*b^4*c^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^9-63104*x^6*a^2*c^8-13888
0*x^4*a^3*c^7-108416*x^2*a^4*c^6-190*x^2*b^8*c^2+25600*x^9*b*c^9+26880*((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^6*b^3*c^6+11520*((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^8*b*c^8+10080*((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^5*c^4+3360*((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^6*c^3+720*((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^7*c^2+90*((
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^8*c+20160*((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*b^4*c^5-277760*x^3*a^3*b*c^6+5
0560*x^3*a^2*b^3*c^5+30384*x^3*a*b^5*c^4-99904*x^2*a^3*b^2*c^5+56424*x^2*a^2*b^4
*c^4-1808*x^2*a*b^6*c^3-108416*x*a^4*b*c^5+38976*x*a^3*b^3*c^4-408*x*a^2*b^5*c^3
-200*x*a*b^7*c^2+2048*x^8*a*c^9+23040*((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^7*b^2*c^7-1616*x^3*b^7*
c^3+59672*x^6*b^4*c^6+57088*x^8*b^2*c^8+1940*x^4*b^6*c^4-10*x*b^9*c+24904*x^5*b^
5*c^5+74752*x^7*b^3*c^7+5120*x^10*c^10)/d^11/(2*c^2*x^3+3*b*c*x^2+2*a*c*x+b^2*x+
a*b)/(2*c*x+b)^9/(4*a*c-b^2)^2/c^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{21}{2}}}\,{d x} \]

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)^(21/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(21/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{c x^{2} + b x + a}}{{\left (1024 \, c^{10} d^{10} x^{10} + 5120 \, b c^{9} d^{10} x^{9} + 11520 \, b^{2} c^{8} d^{10} x^{8} + 15360 \, b^{3} c^{7} d^{10} x^{7} + 13440 \, b^{4} c^{6} d^{10} x^{6} + 8064 \, b^{5} c^{5} d^{10} x^{5} + 3360 \, b^{6} c^{4} d^{10} x^{4} + 960 \, b^{7} c^{3} d^{10} x^{3} + 180 \, b^{8} c^{2} d^{10} x^{2} + 20 \, b^{9} c d^{10} x + b^{10} d^{10}\right )} \sqrt{2 \, c d x + b d}}, x\right ) \]

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)^(21/2),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)/((1024*c^10*d^10*x^10 + 5120*b*c^9*d^10*x^9 + 11520*b^2*c^8*d^10*x^8 +
15360*b^3*c^7*d^10*x^7 + 13440*b^4*c^6*d^10*x^6 + 8064*b^5*c^5*d^10*x^5 + 3360*b
^6*c^4*d^10*x^4 + 960*b^7*c^3*d^10*x^3 + 180*b^8*c^2*d^10*x^2 + 20*b^9*c*d^10*x
+ b^10*d^10)*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)**(21/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{21}{2}}}\,{d x} \]

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)^(21/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(21/2), x)