3.1342 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{9/2}} \, dx\)

Optimal. Leaf size=211 \[ -\frac{5 \left (b^2-4 a c\right )^{5/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{84 c^4 d^{9/2} \sqrt{a+b x+c x^2}}+\frac{5 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{84 c^3 d^5}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{42 c^2 d^3 (b d+2 c d x)^{3/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{7 c d (b d+2 c d x)^{7/2}} \]

[Out]

(5*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(84*c^3*d^5) - (5*(a + b*x + c*x^2
)^(3/2))/(42*c^2*d^3*(b*d + 2*c*d*x)^(3/2)) - (a + b*x + c*x^2)^(5/2)/(7*c*d*(b*
d + 2*c*d*x)^(7/2)) - (5*(b^2 - 4*a*c)^(5/4)*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])/(84*c^4*d^(9/2)*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.507163, antiderivative size = 211, 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 \[ -\frac{5 \left (b^2-4 a c\right )^{5/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{84 c^4 d^{9/2} \sqrt{a+b x+c x^2}}+\frac{5 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{84 c^3 d^5}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{42 c^2 d^3 (b d+2 c d x)^{3/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{7 c d (b d+2 c d x)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(9/2),x]

[Out]

(5*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(84*c^3*d^5) - (5*(a + b*x + c*x^2
)^(3/2))/(42*c^2*d^3*(b*d + 2*c*d*x)^(3/2)) - (a + b*x + c*x^2)^(5/2)/(7*c*d*(b*
d + 2*c*d*x)^(7/2)) - (5*(b^2 - 4*a*c)^(5/4)*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])/(84*c^4*d^(9/2)*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 112.185, size = 202, normalized size = 0.96 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{7 c d \left (b d + 2 c d x\right )^{\frac{7}{2}}} - \frac{5 \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{42 c^{2} d^{3} \left (b d + 2 c d x\right )^{\frac{3}{2}}} + \frac{5 \sqrt{b d + 2 c d x} \sqrt{a + b x + c x^{2}}}{84 c^{3} d^{5}} - \frac{5 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{5}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{84 c^{4} d^{\frac{9}{2}} \sqrt{a + b x + c x^{2}}} \]

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)**(9/2),x)

[Out]

-(a + b*x + c*x**2)**(5/2)/(7*c*d*(b*d + 2*c*d*x)**(7/2)) - 5*(a + b*x + c*x**2)
**(3/2)/(42*c**2*d**3*(b*d + 2*c*d*x)**(3/2)) + 5*sqrt(b*d + 2*c*d*x)*sqrt(a + b
*x + c*x**2)/(84*c**3*d**5) - 5*sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*(-4*a*
c + b**2)**(5/4)*elliptic_f(asin(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(
1/4))), -1)/(84*c**4*d**(9/2)*sqrt(a + b*x + c*x**2))

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Mathematica [C]  time = 0.870404, size = 195, normalized size = 0.92 \[ \frac{-\frac{(b+2 c x) (a+x (b+c x)) \left (-16 \left (b^2-4 a c\right ) (b+2 c x)^2+3 \left (b^2-4 a c\right )^2-7 (b+2 c x)^4\right )}{4 c^3}-\frac{5 i \left (b^2-4 a c\right ) (b+2 c x)^{11/2} \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{c^4 \sqrt{-\sqrt{b^2-4 a c}}}}{84 \sqrt{a+x (b+c x)} (d (b+2 c x))^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(9/2),x]

[Out]

(-((b + 2*c*x)*(a + x*(b + c*x))*(3*(b^2 - 4*a*c)^2 - 16*(b^2 - 4*a*c)*(b + 2*c*
x)^2 - 7*(b + 2*c*x)^4))/(4*c^3) - ((5*I)*(b^2 - 4*a*c)*(b + 2*c*x)^(11/2)*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]]))/(84*(d*(b + 2*c*x))^(9/
2)*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.031, size = 1310, normalized size = 6.2 \[ \text{result too large to display} \]

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)^(9/2),x)

[Out]

1/168*(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^3*a*c^4-40
*((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)*Elliptic
F(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+240*((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*a*b*c^3-60*((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+56*c^6*x^6+120*((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*a*b^2*c^2-30*((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+168*b*c^5*x^5+20*((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)*a*b^3*c-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^5-72*x^4*a
*c^5+228*x^4*b^2*c^4-144*x^3*a*b*c^4+176*b^3*c^3*x^3-152*x^2*a^2*c^4-32*x^2*a*b^
2*c^3+70*x^2*b^4*c^2-152*a^2*b*c^3*x+40*a*b^3*c^2*x+10*b^5*c*x-24*a^3*c^3-20*a^2
*b^2*c^2+10*a*b^4*c)/d^5/(2*c^2*x^3+3*b*c*x^2+2*a*c*x+b^2*x+a*b)/(2*c*x+b)^3/c^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{9}{2}}}\,{d x} \]

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)^(9/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(9/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{c x^{2} + b x + a}}{{\left (16 \, c^{4} d^{4} x^{4} + 32 \, b c^{3} d^{4} x^{3} + 24 \, b^{2} c^{2} d^{4} x^{2} + 8 \, b^{3} c d^{4} x + b^{4} d^{4}\right )} \sqrt{2 \, c d x + b d}}, x\right ) \]

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)^(9/2),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)/((16*c^4*d^4*x^4 + 32*b*c^3*d^4*x^3 + 24*b^2*c^2*d^4*x^2 + 8*b^3*c*d^4*
x + b^4*d^4)*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)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{9}{2}}}\,{d x} \]

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)^(9/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(9/2), x)