3.1382 \(\int \frac{1}{\sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=187 \[ \frac{20 c \sqrt{b d+2 c d x}}{3 d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{b d+2 c d x}}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{40 c \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{3 \sqrt{d} \left (b^2-4 a c\right )^{7/4} \sqrt{a+b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.427063, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{20 c \sqrt{b d+2 c d x}}{3 d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{b d+2 c d x}}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{40 c \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{3 \sqrt{d} \left (b^2-4 a c\right )^{7/4} \sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 100.588, size = 178, normalized size = 0.95 \[ \frac{20 c \sqrt{b d + 2 c d x}}{3 d \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}}} + \frac{40 c \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{3 \sqrt{d} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} \sqrt{a + b x + c x^{2}}} - \frac{2 \sqrt{b d + 2 c d x}}{3 d \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

20*c*sqrt(b*d + 2*c*d*x)/(3*d*(-4*a*c + b**2)**2*sqrt(a + b*x + c*x**2)) + 40*c*
sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*elliptic_f(asin(sqrt(b*d + 2*c*d*x)/(s
qrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(3*sqrt(d)*(-4*a*c + b**2)**(7/4)*sqrt(a +
b*x + c*x**2)) - 2*sqrt(b*d + 2*c*d*x)/(3*d*(-4*a*c + b**2)*(a + b*x + c*x**2)**
(3/2))

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Mathematica [C]  time = 0.839264, size = 168, normalized size = 0.9 \[ \frac{2 \left (-\frac{(b+2 c x) \left (-14 a c+b^2-10 c x (b+c x)\right )}{a+x (b+c x)}+\frac{20 i c (b+2 c x)^{3/2} \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{\sqrt{-\sqrt{b^2-4 a c}}}\right )}{3 \left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-(((b + 2*c*x)*(b^2 - 14*a*c - 10*c*x*(b + c*x)))/(a + x*(b + c*x))) + ((20*
I)*c*(b + 2*c*x)^(3/2)*Sqrt[(c*(a + x*(b + c*x)))/(b + 2*c*x)^2]*EllipticF[I*Arc
Sinh[Sqrt[-Sqrt[b^2 - 4*a*c]]/Sqrt[b + 2*c*x]], -1])/Sqrt[-Sqrt[b^2 - 4*a*c]]))/
(3*(b^2 - 4*a*c)^2*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.054, size = 491, normalized size = 2.6 \[{\frac{2}{3\,d \left ( 2\,cx+b \right ) \left ( 4\,ac-{b}^{2} \right ) ^{2}}\sqrt{d \left ( 2\,cx+b \right ) } \left ( 10\,{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){x}^{2}{c}^{2}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-4\,ac+{b}^{2}}\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}+10\,{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) xbc\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-4\,ac+{b}^{2}}\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}+10\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}ac+20\,{c}^{3}{x}^{3}+30\,b{c}^{2}{x}^{2}+28\,a{c}^{2}x+8\,x{b}^{2}c+14\,abc-{b}^{3} \right ) \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(d*(2*c*x+b))^(1/2)*(10*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))*x^2*c^2*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*
c+b^2)^(1/2))^(1/2)*(-4*a*c+b^2)^(1/2)*((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)+10*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))*x*b*c*((-b-2*c*x+
(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-4*a*c+b^2)^(1/2)*((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)+1
0*((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)*Ellipti
cF(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*c+20*c^3*x^3+30*b*c^2*x^2+28*a*c^2*x+8*x*b^2*c+14*a*b*c-b^3
)/d/(2*c*x+b)/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*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. \[ \int \frac{1}{\sqrt{d \left (b + 2 c x\right )} \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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