3.122 \(\int \frac{x \left (d+e x^2\right )}{\sqrt{a x+b x^3+c x^5}} \, dx\)

Optimal. Leaf size=287 \[ \frac{2 d x^2 \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{3}{4};\frac{1}{2},\frac{1}{2};\frac{7}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{3 \sqrt{a x+b x^3+c x^5}}+\frac{2 e x^4 \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{7}{4};\frac{1}{2},\frac{1}{2};\frac{11}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{7 \sqrt{a x+b x^3+c x^5}} \]

[Out]

(2*d*x^2*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b + Sqr
t[b^2 - 4*a*c])]*AppellF1[3/4, 1/2, 1/2, 7/4, (-2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])
, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])])/(3*Sqrt[a*x + b*x^3 + c*x^5]) + (2*e*x^4*
Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b + Sqrt[b^2 - 4
*a*c])]*AppellF1[7/4, 1/2, 1/2, 11/4, (-2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*
x^2)/(b + Sqrt[b^2 - 4*a*c])])/(7*Sqrt[a*x + b*x^3 + c*x^5])

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Rubi [A]  time = 1.0874, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{2 d x^2 \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{3}{4};\frac{1}{2},\frac{1}{2};\frac{7}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{3 \sqrt{a x+b x^3+c x^5}}+\frac{2 e x^4 \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{7}{4};\frac{1}{2},\frac{1}{2};\frac{11}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{7 \sqrt{a x+b x^3+c x^5}} \]

Antiderivative was successfully verified.

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

[Out]

(2*d*x^2*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b + Sqr
t[b^2 - 4*a*c])]*AppellF1[3/4, 1/2, 1/2, 7/4, (-2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])
, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])])/(3*Sqrt[a*x + b*x^3 + c*x^5]) + (2*e*x^4*
Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b + Sqrt[b^2 - 4
*a*c])]*AppellF1[7/4, 1/2, 1/2, 11/4, (-2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*
x^2)/(b + Sqrt[b^2 - 4*a*c])])/(7*Sqrt[a*x + b*x^3 + c*x^5])

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Rubi in Sympy [A]  time = 91.4364, size = 280, normalized size = 0.98 \[ \frac{2 d x^{2} \left (a + b x^{2} + c x^{4}\right ) \operatorname{appellf_{1}}{\left (\frac{3}{4},\frac{1}{2},\frac{1}{2},\frac{7}{4},- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{3 a \sqrt{\frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{a x + b x^{3} + c x^{5}}} + \frac{2 e x^{4} \left (a + b x^{2} + c x^{4}\right ) \operatorname{appellf_{1}}{\left (\frac{7}{4},\frac{1}{2},\frac{1}{2},\frac{11}{4},- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{7 a \sqrt{\frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{a x + b x^{3} + c x^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*d*x**2*(a + b*x**2 + c*x**4)*appellf1(3/4, 1/2, 1/2, 7/4, -2*c*x**2/(b - sqrt(
-4*a*c + b**2)), -2*c*x**2/(b + sqrt(-4*a*c + b**2)))/(3*a*sqrt(2*c*x**2/(b - sq
rt(-4*a*c + b**2)) + 1)*sqrt(2*c*x**2/(b + sqrt(-4*a*c + b**2)) + 1)*sqrt(a*x +
b*x**3 + c*x**5)) + 2*e*x**4*(a + b*x**2 + c*x**4)*appellf1(7/4, 1/2, 1/2, 11/4,
-2*c*x**2/(b - sqrt(-4*a*c + b**2)), -2*c*x**2/(b + sqrt(-4*a*c + b**2)))/(7*a*
sqrt(2*c*x**2/(b - sqrt(-4*a*c + b**2)) + 1)*sqrt(2*c*x**2/(b + sqrt(-4*a*c + b*
*2)) + 1)*sqrt(a*x + b*x**3 + c*x**5))

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Mathematica [B]  time = 0.754711, size = 639, normalized size = 2.23 \[ \frac{a x^3 \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (-\frac{49 d F_1\left (\frac{3}{4};\frac{1}{2},\frac{1}{2};\frac{7}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )}{x^2 \left (\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (\frac{7}{4};\frac{1}{2},\frac{3}{2};\frac{11}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{7}{4};\frac{3}{2},\frac{1}{2};\frac{11}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )-7 a F_1\left (\frac{3}{4};\frac{1}{2},\frac{1}{2};\frac{7}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )}-\frac{33 e x^2 F_1\left (\frac{7}{4};\frac{1}{2},\frac{1}{2};\frac{11}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )}{x^2 \left (\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (\frac{11}{4};\frac{1}{2},\frac{3}{2};\frac{15}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{11}{4};\frac{3}{2},\frac{1}{2};\frac{15}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )-11 a F_1\left (\frac{7}{4};\frac{1}{2},\frac{1}{2};\frac{11}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )}\right )}{42 c \left (x \left (a+b x^2+c x^4\right )\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(a*x^3*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)*((-49
*d*AppellF1[3/4, 1/2, 1/2, 7/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-
b + Sqrt[b^2 - 4*a*c])])/(-7*a*AppellF1[3/4, 1/2, 1/2, 7/4, (-2*c*x^2)/(b + Sqrt
[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + x^2*((b + Sqrt[b^2 - 4*a*c
])*AppellF1[7/4, 1/2, 3/2, 11/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(
-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4*a*c])*AppellF1[7/4, 3/2, 1/2, 11/4,
(-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])) - (33
*e*x^2*AppellF1[7/4, 1/2, 1/2, 11/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^
2)/(-b + Sqrt[b^2 - 4*a*c])])/(-11*a*AppellF1[7/4, 1/2, 1/2, 11/4, (-2*c*x^2)/(b
+ Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + x^2*((b + Sqrt[b^2
- 4*a*c])*AppellF1[11/4, 1/2, 3/2, 15/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*
c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4*a*c])*AppellF1[11/4, 3/2, 1
/2, 15/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])
]))))/(42*c*(x*(a + b*x^2 + c*x^4))^(3/2))

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Maple [F]  time = 0.032, size = 0, normalized size = 0. \[ \int{x \left ( e{x}^{2}+d \right ){\frac{1}{\sqrt{c{x}^{5}+b{x}^{3}+ax}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e*x^3 + d*x)/sqrt(c*x^5 + b*x^3 + a*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x \left (d + e x^{2}\right )}{\sqrt{x \left (a + b x^{2} + c x^{4}\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*(d + e*x**2)/sqrt(x*(a + b*x**2 + c*x**4)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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