3.1350 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{11/2}} \, dx\)

Optimal. Leaf size=310 \[ -\frac{\left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{12 c^4 d^{11/2} \sqrt{a+b x+c x^2}}+\frac{\left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{12 c^4 d^{11/2} \sqrt{a+b x+c x^2}}-\frac{\sqrt{a+b x+c x^2}}{12 c^3 d^5 \sqrt{b d+2 c d x}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{18 c^2 d^3 (b d+2 c d x)^{5/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{9 c d (b d+2 c d x)^{9/2}} \]

[Out]

-Sqrt[a + b*x + c*x^2]/(12*c^3*d^5*Sqrt[b*d + 2*c*d*x]) - (a + b*x + c*x^2)^(3/2
)/(18*c^2*d^3*(b*d + 2*c*d*x)^(5/2)) - (a + b*x + c*x^2)^(5/2)/(9*c*d*(b*d + 2*c
*d*x)^(9/2)) + ((b^2 - 4*a*c)^(3/4)*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]
*EllipticE[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(12*c
^4*d^(11/2)*Sqrt[a + b*x + c*x^2]) - ((b^2 - 4*a*c)^(3/4)*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])/(12*c^4*d^(11/2)*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.917426, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{12 c^4 d^{11/2} \sqrt{a+b x+c x^2}}+\frac{\left (b^2-4 a c\right )^{3/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{12 c^4 d^{11/2} \sqrt{a+b x+c x^2}}-\frac{\sqrt{a+b x+c x^2}}{12 c^3 d^5 \sqrt{b d+2 c d x}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{18 c^2 d^3 (b d+2 c d x)^{5/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{9 c d (b d+2 c d x)^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(11/2),x]

[Out]

-Sqrt[a + b*x + c*x^2]/(12*c^3*d^5*Sqrt[b*d + 2*c*d*x]) - (a + b*x + c*x^2)^(3/2
)/(18*c^2*d^3*(b*d + 2*c*d*x)^(5/2)) - (a + b*x + c*x^2)^(5/2)/(9*c*d*(b*d + 2*c
*d*x)^(9/2)) + ((b^2 - 4*a*c)^(3/4)*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]
*EllipticE[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(12*c
^4*d^(11/2)*Sqrt[a + b*x + c*x^2]) - ((b^2 - 4*a*c)^(3/4)*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])/(12*c^4*d^(11/2)*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 170.035, size = 292, normalized size = 0.94 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{9 c d \left (b d + 2 c d x\right )^{\frac{9}{2}}} - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{18 c^{2} d^{3} \left (b d + 2 c d x\right )^{\frac{5}{2}}} - \frac{\sqrt{a + b x + c x^{2}}}{12 c^{3} d^{5} \sqrt{b d + 2 c d x}} + \frac{\sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{12 c^{4} d^{\frac{11}{2}} \sqrt{a + b x + c x^{2}}} - \frac{\sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{12 c^{4} d^{\frac{11}{2}} \sqrt{a + b x + c x^{2}}} \]

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)**(11/2),x)

[Out]

-(a + b*x + c*x**2)**(5/2)/(9*c*d*(b*d + 2*c*d*x)**(9/2)) - (a + b*x + c*x**2)**
(3/2)/(18*c**2*d**3*(b*d + 2*c*d*x)**(5/2)) - sqrt(a + b*x + c*x**2)/(12*c**3*d*
*5*sqrt(b*d + 2*c*d*x)) + sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*(-4*a*c + b*
*2)**(3/4)*elliptic_e(asin(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4)))
, -1)/(12*c**4*d**(11/2)*sqrt(a + b*x + c*x**2)) - sqrt(c*(a + b*x + c*x**2)/(4*
a*c - b**2))*(-4*a*c + b**2)**(3/4)*elliptic_f(asin(sqrt(b*d + 2*c*d*x)/(sqrt(d)
*(-4*a*c + b**2)**(1/4))), -1)/(12*c**4*d**(11/2)*sqrt(a + b*x + c*x**2))

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Mathematica [C]  time = 1.35914, size = 251, normalized size = 0.81 \[ \frac{-\frac{(b+2 c x) (a+x (b+c x)) \left (4 c^2 \left (a^2+4 a c x^2+15 c^2 x^4\right )+2 b^2 c \left (a+43 c x^2\right )+8 b c^2 x \left (2 a+15 c x^2\right )+3 b^4+26 b^3 c x\right )}{3 c^3}+\frac{i (b+2 c x)^7 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{c^4 \left (-\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )^{3/2}}}{12 \sqrt{a+x (b+c x)} (d (b+2 c x))^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(11/2),x]

[Out]

(-((b + 2*c*x)*(a + x*(b + c*x))*(3*b^4 + 26*b^3*c*x + 8*b*c^2*x*(2*a + 15*c*x^2
) + 2*b^2*c*(a + 43*c*x^2) + 4*c^2*(a^2 + 4*a*c*x^2 + 15*c^2*x^4)))/(3*c^3) + (I
*(b + 2*c*x)^7*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*(EllipticE[I*ArcSinh[S
qrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]], -1] - EllipticF[I*ArcSinh[Sqrt[-((b + 2*
c*x)/Sqrt[b^2 - 4*a*c])]], -1]))/(c^4*(-((b + 2*c*x)/Sqrt[b^2 - 4*a*c]))^(3/2)))
/(12*(d*(b + 2*c*x))^(11/2)*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.043, size = 1489, normalized size = 4.8 \[ \text{result too large to display} \]

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)^(11/2),x)

[Out]

1/72*(c*x^2+b*x+a)^(1/2)*(d*(2*c*x+b))^(1/2)*(288*EllipticE(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))*x^2*a*b^2*c^3*((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)+384*EllipticE(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))*x^3*a*b
*c^4*((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)+96*E
llipticE(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))*x*a*b^3*c^2*((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)-58*x^2*b^4*c^2-6*b^5*c*x+12*((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)*EllipticE(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))*a*b^4*c-304*x^3*a*b*c^4-208*x^2*a*b^2*c^3-40*
a^2*b*c^3*x-8*a^3*c^3-3*((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)*EllipticE(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))*b^6+192*EllipticE(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))*x^4*a*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)-48*EllipticE(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))*x^4*b^2*c^4*((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)-360*b*c^5*x^5-152*x^4*a*c^5-4
12*x^4*b^2*c^4-224*b^3*c^3*x^3-40*x^2*a^2*c^4-56*a*b^3*c^2*x-120*c^6*x^6-6*a*b^4
*c-4*a^2*b^2*c^2-96*EllipticE(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))*x^3*b^3*c^3*((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)-72*EllipticE(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))*x^2*b^4*c^2*((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)-24*EllipticE(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))*x*b^5*c*((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))/d^6/(2*c^2*x^3+3*b*c*x^2+2*
a*c*x+b^2*x+a*b)/(2*c*x+b)^4/c^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{11}{2}}}\,{d x} \]

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)^(11/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(11/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{c x^{2} + b x + a}}{{\left (32 \, c^{5} d^{5} x^{5} + 80 \, b c^{4} d^{5} x^{4} + 80 \, b^{2} c^{3} d^{5} x^{3} + 40 \, b^{3} c^{2} d^{5} x^{2} + 10 \, b^{4} c d^{5} x + b^{5} d^{5}\right )} \sqrt{2 \, c d x + b d}}, x\right ) \]

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)^(11/2),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)/((32*c^5*d^5*x^5 + 80*b*c^4*d^5*x^4 + 80*b^2*c^3*d^5*x^3 + 40*b^3*c^2*d
^5*x^2 + 10*b^4*c*d^5*x + b^5*d^5)*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)**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{11}{2}}}\,{d x} \]

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)^(11/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(11/2), x)