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

Optimal. Leaf size=425 \[ -\frac{\sqrt [4]{a} \sqrt{x} \left (\sqrt{a} \sqrt{c} \left (b^2-20 a c\right )+2 b \left (b^2-8 a c\right )\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{70 c^{7/4} \sqrt{a x+b x^3+c x^5}}+\frac{2 \sqrt [4]{a} b \sqrt{x} \left (b^2-8 a c\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{35 c^{7/4} \sqrt{a x+b x^3+c x^5}}-\frac{2 b x^{3/2} \left (b^2-8 a c\right ) \left (a+b x^2+c x^4\right )}{35 c^{3/2} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{a x+b x^3+c x^5}}+\frac{\sqrt{x} \left (10 a c+b^2+3 b c x^2\right ) \sqrt{a x+b x^3+c x^5}}{35 c}+\frac{\left (a x+b x^3+c x^5\right )^{3/2}}{7 \sqrt{x}} \]

[Out]

(-2*b*(b^2 - 8*a*c)*x^(3/2)*(a + b*x^2 + c*x^4))/(35*c^(3/2)*(Sqrt[a] + Sqrt[c]*
x^2)*Sqrt[a*x + b*x^3 + c*x^5]) + (Sqrt[x]*(b^2 + 10*a*c + 3*b*c*x^2)*Sqrt[a*x +
b*x^3 + c*x^5])/(35*c) + (a*x + b*x^3 + c*x^5)^(3/2)/(7*Sqrt[x]) + (2*a^(1/4)*b
*(b^2 - 8*a*c)*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a]
+ Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt
[c]))/4])/(35*c^(7/4)*Sqrt[a*x + b*x^3 + c*x^5]) - (a^(1/4)*(Sqrt[a]*Sqrt[c]*(b^
2 - 20*a*c) + 2*b*(b^2 - 8*a*c))*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2
+ c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2
- b/(Sqrt[a]*Sqrt[c]))/4])/(70*c^(7/4)*Sqrt[a*x + b*x^3 + c*x^5])

_______________________________________________________________________________________

Rubi [A]  time = 0.796389, antiderivative size = 425, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\sqrt [4]{a} \sqrt{x} \left (\sqrt{a} \sqrt{c} \left (b^2-20 a c\right )+2 b \left (b^2-8 a c\right )\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{70 c^{7/4} \sqrt{a x+b x^3+c x^5}}+\frac{2 \sqrt [4]{a} b \sqrt{x} \left (b^2-8 a c\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{35 c^{7/4} \sqrt{a x+b x^3+c x^5}}-\frac{2 b x^{3/2} \left (b^2-8 a c\right ) \left (a+b x^2+c x^4\right )}{35 c^{3/2} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{a x+b x^3+c x^5}}+\frac{\sqrt{x} \left (10 a c+b^2+3 b c x^2\right ) \sqrt{a x+b x^3+c x^5}}{35 c}+\frac{\left (a x+b x^3+c x^5\right )^{3/2}}{7 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*b*(b^2 - 8*a*c)*x^(3/2)*(a + b*x^2 + c*x^4))/(35*c^(3/2)*(Sqrt[a] + Sqrt[c]*
x^2)*Sqrt[a*x + b*x^3 + c*x^5]) + (Sqrt[x]*(b^2 + 10*a*c + 3*b*c*x^2)*Sqrt[a*x +
b*x^3 + c*x^5])/(35*c) + (a*x + b*x^3 + c*x^5)^(3/2)/(7*Sqrt[x]) + (2*a^(1/4)*b
*(b^2 - 8*a*c)*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a]
+ Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt
[c]))/4])/(35*c^(7/4)*Sqrt[a*x + b*x^3 + c*x^5]) - (a^(1/4)*(Sqrt[a]*Sqrt[c]*(b^
2 - 20*a*c) + 2*b*(b^2 - 8*a*c))*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2
+ c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2
- b/(Sqrt[a]*Sqrt[c]))/4])/(70*c^(7/4)*Sqrt[a*x + b*x^3 + c*x^5])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 96.3897, size = 394, normalized size = 0.93 \[ \frac{2 \sqrt [4]{a} b \sqrt{x} \sqrt{\frac{a + b x^{2} + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (- 8 a c + b^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{35 c^{\frac{7}{4}} \sqrt{a x + b x^{3} + c x^{5}}} - \frac{\sqrt [4]{a} \sqrt{x} \sqrt{\frac{a + b x^{2} + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (\sqrt{a} \sqrt{c} \left (- 20 a c + b^{2}\right ) + 2 b \left (- 8 a c + b^{2}\right )\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{70 c^{\frac{7}{4}} \sqrt{a x + b x^{3} + c x^{5}}} - \frac{2 b x^{\frac{3}{2}} \left (- 8 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )}{35 c^{\frac{3}{2}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \sqrt{a x + b x^{3} + c x^{5}}} + \frac{\left (a x + b x^{3} + c x^{5}\right )^{\frac{3}{2}}}{7 \sqrt{x}} + \frac{\sqrt{x} \left (10 a c + b^{2} + 3 b c x^{2}\right ) \sqrt{a x + b x^{3} + c x^{5}}}{35 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**(1/4)*b*sqrt(x)*sqrt((a + b*x**2 + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sq
rt(a) + sqrt(c)*x**2)*(-8*a*c + b**2)*elliptic_e(2*atan(c**(1/4)*x/a**(1/4)), 1/
2 - b/(4*sqrt(a)*sqrt(c)))/(35*c**(7/4)*sqrt(a*x + b*x**3 + c*x**5)) - a**(1/4)*
sqrt(x)*sqrt((a + b*x**2 + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sqrt(
c)*x**2)*(sqrt(a)*sqrt(c)*(-20*a*c + b**2) + 2*b*(-8*a*c + b**2))*elliptic_f(2*a
tan(c**(1/4)*x/a**(1/4)), 1/2 - b/(4*sqrt(a)*sqrt(c)))/(70*c**(7/4)*sqrt(a*x + b
*x**3 + c*x**5)) - 2*b*x**(3/2)*(-8*a*c + b**2)*(a + b*x**2 + c*x**4)/(35*c**(3/
2)*(sqrt(a) + sqrt(c)*x**2)*sqrt(a*x + b*x**3 + c*x**5)) + (a*x + b*x**3 + c*x**
5)**(3/2)/(7*sqrt(x)) + sqrt(x)*(10*a*c + b**2 + 3*b*c*x**2)*sqrt(a*x + b*x**3 +
c*x**5)/(35*c)

_______________________________________________________________________________________

Mathematica [C]  time = 3.09902, size = 540, normalized size = 1.27 \[ \frac{\sqrt{x} \left (2 c x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left (15 a^2 c+a \left (b^2+23 b c x^2+20 c^2 x^4\right )+x^2 \left (b^3+9 b^2 c x^2+13 b c^2 x^4+5 c^3 x^6\right )\right )+i \left (-20 a^2 c^2+9 a b^2 c-8 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}-b^4\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )-i b \left (b^2-8 a c\right ) \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )\right )}{70 c^2 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{x \left (a+b x^2+c x^4\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*(2*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(15*a^2*c + a*(b^2 + 23*b*c*x^2
+ 20*c^2*x^4) + x^2*(b^3 + 9*b^2*c*x^2 + 13*b*c^2*x^4 + 5*c^3*x^6)) - I*b*(b^2 -
8*a*c)*(-b + Sqrt[b^2 - 4*a*c])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqr
t[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*
c])]*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b
^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + I*(-b^4 + 9*a*b^2*c - 20*a^2*c^2 + b^3*S
qrt[b^2 - 4*a*c] - 8*a*b*c*Sqrt[b^2 - 4*a*c])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*
x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sq
rt[b^2 - 4*a*c])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x]
, (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]))/(70*c^2*Sqrt[c/(b + Sqrt[b^
2 - 4*a*c])]*Sqrt[x*(a + b*x^2 + c*x^4)])

_______________________________________________________________________________________

Maple [B]  time = 0.029, size = 1394, normalized size = 3.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/70*(x*(c*x^4+b*x^2+a))^(1/2)*(-10*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*x^9*b*c^3
-10*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(-4*a*c+b^2)^(1/2)*x^9*c^3-26*((-b+(-4*a*c
+b^2)^(1/2))/a)^(1/2)*x^7*b^2*c^2-26*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(-4*a*c+b
^2)^(1/2)*x^7*b*c^2-40*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*x^5*a*b*c^2-40*((-b+(-4
*a*c+b^2)^(1/2))/a)^(1/2)*(-4*a*c+b^2)^(1/2)*x^5*a*c^2-18*((-b+(-4*a*c+b^2)^(1/2
))/a)^(1/2)*x^5*b^3*c-18*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(-4*a*c+b^2)^(1/2)*x^
5*b^2*c-46*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*x^3*a*b^2*c-46*((-b+(-4*a*c+b^2)^(1
/2))/a)^(1/2)*(-4*a*c+b^2)^(1/2)*x^3*a*b*c-2*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*x
^3*b^4-2*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(-4*a*c+b^2)^(1/2)*x^3*b^3+12*(-2*(x^
2*(-4*a*c+b^2)^(1/2)-b*x^2-2*a)/a)^(1/2)*((x^2*(-4*a*c+b^2)^(1/2)+b*x^2+2*a)/a)^
(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*2^(1/2)*((b*
(-4*a*c+b^2)^(1/2)-2*a*c+b^2)/a/c)^(1/2))*a^2*b*c-20*(-2*(x^2*(-4*a*c+b^2)^(1/2)
-b*x^2-2*a)/a)^(1/2)*((x^2*(-4*a*c+b^2)^(1/2)+b*x^2+2*a)/a)^(1/2)*EllipticF(1/2*
x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*2^(1/2)*((b*(-4*a*c+b^2)^(1/2)-2
*a*c+b^2)/a/c)^(1/2))*(-4*a*c+b^2)^(1/2)*a^2*c-3*(-2*(x^2*(-4*a*c+b^2)^(1/2)-b*x
^2-2*a)/a)^(1/2)*((x^2*(-4*a*c+b^2)^(1/2)+b*x^2+2*a)/a)^(1/2)*EllipticF(1/2*x*2^
(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*2^(1/2)*((b*(-4*a*c+b^2)^(1/2)-2*a*c
+b^2)/a/c)^(1/2))*a*b^3+(-2*(x^2*(-4*a*c+b^2)^(1/2)-b*x^2-2*a)/a)^(1/2)*((x^2*(-
4*a*c+b^2)^(1/2)+b*x^2+2*a)/a)^(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(
1/2))/a)^(1/2),1/2*2^(1/2)*((b*(-4*a*c+b^2)^(1/2)-2*a*c+b^2)/a/c)^(1/2))*(-4*a*c
+b^2)^(1/2)*a*b^2-32*(-2*(x^2*(-4*a*c+b^2)^(1/2)-b*x^2-2*a)/a)^(1/2)*((x^2*(-4*a
*c+b^2)^(1/2)+b*x^2+2*a)/a)^(1/2)*EllipticE(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2
))/a)^(1/2),1/2*2^(1/2)*((b*(-4*a*c+b^2)^(1/2)-2*a*c+b^2)/a/c)^(1/2))*a^2*b*c+4*
(-2*(x^2*(-4*a*c+b^2)^(1/2)-b*x^2-2*a)/a)^(1/2)*((x^2*(-4*a*c+b^2)^(1/2)+b*x^2+2
*a)/a)^(1/2)*EllipticE(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*2^(1/
2)*((b*(-4*a*c+b^2)^(1/2)-2*a*c+b^2)/a/c)^(1/2))*a*b^3-30*((-b+(-4*a*c+b^2)^(1/2
))/a)^(1/2)*x*a^2*b*c-30*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(-4*a*c+b^2)^(1/2)*x*
a^2*c-2*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*x*a*b^3-2*((-b+(-4*a*c+b^2)^(1/2))/a)^
(1/2)*(-4*a*c+b^2)^(1/2)*x*a*b^2)/x^(1/2)/(c*x^4+b*x^2+a)/c/((-b+(-4*a*c+b^2)^(1
/2))/a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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**5+b*x**3+a*x)**(3/2)/x**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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