3.1373 \(\int \frac{(b d+2 c d x)^{9/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=258 \[ -\frac{84 d^{9/2} \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 \sqrt{a+b x+c x^2}}+\frac{84 d^{9/2} \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 \sqrt{a+b x+c x^2}}+\frac{56}{5} c d^3 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}-\frac{2 d (b d+2 c d x)^{7/2}}{\sqrt{a+b x+c x^2}} \]

[Out]

(-2*d*(b*d + 2*c*d*x)^(7/2))/Sqrt[a + b*x + c*x^2] + (56*c*d^3*(b*d + 2*c*d*x)^(
3/2)*Sqrt[a + b*x + c*x^2])/5 + (84*(b^2 - 4*a*c)^(7/4)*d^(9/2)*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])/(5*Sqrt[a + b*x + c*x^2]) - (84*(b^2 - 4*a*c)^(7/4)*d^(9/
2)*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])/(5*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.827572, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{84 d^{9/2} \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 \sqrt{a+b x+c x^2}}+\frac{84 d^{9/2} \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 \sqrt{a+b x+c x^2}}+\frac{56}{5} c d^3 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}-\frac{2 d (b d+2 c d x)^{7/2}}{\sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2)^(3/2),x]

[Out]

(-2*d*(b*d + 2*c*d*x)^(7/2))/Sqrt[a + b*x + c*x^2] + (56*c*d^3*(b*d + 2*c*d*x)^(
3/2)*Sqrt[a + b*x + c*x^2])/5 + (84*(b^2 - 4*a*c)^(7/4)*d^(9/2)*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])/(5*Sqrt[a + b*x + c*x^2]) - (84*(b^2 - 4*a*c)^(7/4)*d^(9/
2)*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])/(5*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 142.218, size = 252, normalized size = 0.98 \[ \frac{56 c d^{3} \left (b d + 2 c d x\right )^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}}{5} + \frac{84 d^{\frac{9}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{5 \sqrt{a + b x + c x^{2}}} - \frac{84 d^{\frac{9}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{5 \sqrt{a + b x + c x^{2}}} - \frac{2 d \left (b d + 2 c d x\right )^{\frac{7}{2}}}{\sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**(9/2)/(c*x**2+b*x+a)**(3/2),x)

[Out]

56*c*d**3*(b*d + 2*c*d*x)**(3/2)*sqrt(a + b*x + c*x**2)/5 + 84*d**(9/2)*sqrt(c*(
a + b*x + c*x**2)/(4*a*c - b**2))*(-4*a*c + b**2)**(7/4)*elliptic_e(asin(sqrt(b*
d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(5*sqrt(a + b*x + c*x**2)) -
84*d**(9/2)*sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*(-4*a*c + b**2)**(7/4)*el
liptic_f(asin(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(5*sqrt
(a + b*x + c*x**2)) - 2*d*(b*d + 2*c*d*x)**(7/2)/sqrt(a + b*x + c*x**2)

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Mathematica [C]  time = 2.24461, size = 258, normalized size = 1. \[ -\frac{2 d^5 \sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}} \left (\frac{(b+2 c x)^2 \left (-4 c \left (7 a+2 c x^2\right )+5 b^2-8 b c x\right )}{\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}}+42 i \left (b^2-4 a c\right )^2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-42 i \left (b^2-4 a c\right )^2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{5 \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2)^(3/2),x]

[Out]

(-2*d^5*Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]*(((b + 2*c*x)^2*(5*b^2 - 8*b*c*x
- 4*c*(7*a + 2*c*x^2)))/Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])] - (42*I)*(b^2 - 4
*a*c)^2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*EllipticE[I*ArcSinh[Sqrt[-((b
+ 2*c*x)/Sqrt[b^2 - 4*a*c])]], -1] + (42*I)*(b^2 - 4*a*c)^2*Sqrt[(c*(a + x*(b +
c*x)))/(-b^2 + 4*a*c)]*EllipticF[I*ArcSinh[Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c]
)]], -1]))/(5*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.062, size = 498, normalized size = 1.9 \[ -{\frac{2\,{d}^{4}}{10\,{x}^{3}{c}^{2}+15\,{x}^{2}bc+10\,acx+5\,{b}^{2}x+5\,ab} \left ( 336\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){a}^{2}{c}^{2}-168\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) a{b}^{2}c+21\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){b}^{4}-32\,{c}^{4}{x}^{4}-64\,b{c}^{3}{x}^{3}-112\,{x}^{2}a{c}^{3}-20\,{x}^{2}{b}^{2}{c}^{2}-112\,xab{c}^{2}+12\,{b}^{3}cx-28\,ac{b}^{2}+5\,{b}^{4} \right ) \sqrt{c{x}^{2}+bx+a}\sqrt{d \left ( 2\,cx+b \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a)^(3/2),x)

[Out]

-2/5*d^4*(336*((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^2*c^2-168*((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^2*c+21*((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^4-32*c^4*x^4-64*b*c^3*x^3-112*x^2*a*c^3-20*x^2*
b^2*c^2-112*x*a*b*c^2+12*b^3*c*x-28*a*c*b^2+5*b^4)*(c*x^2+b*x+a)^(1/2)*(d*(2*c*x
+b))^(1/2)/(2*c^2*x^3+3*b*c*x^2+2*a*c*x+b^2*x+a*b)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{9}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(9/2)/(c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")

[Out]

integrate((2*c*d*x + b*d)^(9/2)/(c*x^2 + b*x + a)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(9/2)/(c*x^2 + b*x + a)^(3/2),x, algorithm="fricas")

[Out]

integral((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)/(c*x^2 + b*x + a)^(3/2), 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((2*c*d*x+b*d)**(9/2)/(c*x**2+b*x+a)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{9}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(9/2)/(c*x^2 + b*x + a)^(3/2),x, algorithm="giac")

[Out]

integrate((2*c*d*x + b*d)^(9/2)/(c*x^2 + b*x + a)^(3/2), x)