3.1372 \(\int \frac{1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=184 \[ -\frac{20 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{3 d^{5/2} \left (b^2-4 a c\right )^{7/4} \sqrt{a+b x+c x^2}}-\frac{40 c \sqrt{a+b x+c x^2}}{3 d \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}-\frac{2}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}} \]

[Out]

-2/((b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2]) - (40*c*Sqrt[a
+ b*x + c*x^2])/(3*(b^2 - 4*a*c)^2*d*(b*d + 2*c*d*x)^(3/2)) - (20*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])/(3*(b^2 - 4*a*c)^(7/4)*d^(5/2)*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.429715, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{20 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{3 d^{5/2} \left (b^2-4 a c\right )^{7/4} \sqrt{a+b x+c x^2}}-\frac{40 c \sqrt{a+b x+c x^2}}{3 d \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}-\frac{2}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(3/2)),x]

[Out]

-2/((b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2]) - (40*c*Sqrt[a
+ b*x + c*x^2])/(3*(b^2 - 4*a*c)^2*d*(b*d + 2*c*d*x)^(3/2)) - (20*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])/(3*(b^2 - 4*a*c)^(7/4)*d^(5/2)*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 101.723, size = 177, normalized size = 0.96 \[ - \frac{40 c \sqrt{a + b x + c x^{2}}}{3 d \left (- 4 a c + b^{2}\right )^{2} \left (b d + 2 c d x\right )^{\frac{3}{2}}} - \frac{2}{d \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}} - \frac{20 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{3 d^{\frac{5}{2}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*c*d*x+b*d)**(5/2)/(c*x**2+b*x+a)**(3/2),x)

[Out]

-40*c*sqrt(a + b*x + c*x**2)/(3*d*(-4*a*c + b**2)**2*(b*d + 2*c*d*x)**(3/2)) - 2
/(d*(-4*a*c + b**2)*(b*d + 2*c*d*x)**(3/2)*sqrt(a + b*x + c*x**2)) - 20*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)/(3*d**(5/2)*(-4*a*c + b**2)**(7/4)*sqrt(a + b*x + c
*x**2))

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Mathematica [C]  time = 0.66817, size = 177, normalized size = 0.96 \[ -\frac{2 \left (\sqrt{-\sqrt{b^2-4 a c}} \left (4 c \left (2 a+5 c x^2\right )+3 b^2+20 b c x\right )+10 i (b+2 c x)^{5/2} \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )\right )}{3 d \sqrt{-\sqrt{b^2-4 a c}} \left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)} (d (b+2 c x))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(3/2)),x]

[Out]

(-2*(Sqrt[-Sqrt[b^2 - 4*a*c]]*(3*b^2 + 20*b*c*x + 4*c*(2*a + 5*c*x^2)) + (10*I)*
(b + 2*c*x)^(5/2)*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]))/(3*Sqrt[-Sqrt[b^2 - 4*a*c]]*(b^
2 - 4*a*c)^2*d*(d*(b + 2*c*x))^(3/2)*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.046, size = 365, normalized size = 2. \[ -{\frac{2}{3\,{d}^{3} \left ( 2\,{x}^{3}{c}^{2}+3\,{x}^{2}bc+2\,acx+{b}^{2}x+ab \right ) \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( 2\,cx+b \right ) }\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a} \left ( 10\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}xc+5\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}b+20\,{c}^{2}{x}^{2}+20\,bxc+8\,ac+3\,{b}^{2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^(3/2),x)

[Out]

-2/3*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*(10*((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*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+20*c^2*x^2+20*b*x*c+8*a*c+3*b^2)/d^3/(2*c^2*x^3+3*b*c*x^2+2*a*c*x+b^2
*x+a*b)/(4*a*c-b^2)^2/(2*c*x+b)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a)^(3/2)),x, algorithm="maxima")

[Out]

integrate(1/((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (4 \, c^{3} d^{2} x^{4} + 8 \, b c^{2} d^{2} x^{3} + a b^{2} d^{2} +{\left (5 \, b^{2} c + 4 \, a c^{2}\right )} d^{2} x^{2} +{\left (b^{3} + 4 \, a b c\right )} d^{2} x\right )} \sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a)^(3/2)),x, algorithm="fricas")

[Out]

integral(1/((4*c^3*d^2*x^4 + 8*b*c^2*d^2*x^3 + a*b^2*d^2 + (5*b^2*c + 4*a*c^2)*d
^2*x^2 + (b^3 + 4*a*b*c)*d^2*x)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d \left (b + 2 c x\right )\right )^{\frac{5}{2}} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2*c*d*x+b*d)**(5/2)/(c*x**2+b*x+a)**(3/2),x)

[Out]

Integral(1/((d*(b + 2*c*x))**(5/2)*(a + b*x + c*x**2)**(3/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a)^(3/2)),x, algorithm="giac")

[Out]

integrate(1/((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a)^(3/2)), x)