3.2.38 \(\int \frac {3 a+b x^2}{(a-b x^2)^{7/3}} \, dx\) [138]

3.2.38.1 Optimal result

Integrand size = 22, antiderivative size = 590 \[ \int \frac {3 a+b x^2}{\left (a-b x^2\right )^{7/3}} \, dx=\frac {3 x}{2 \left (a-b x^2\right )^{4/3}}+\frac {9 x}{4 a \sqrt [3]{a-b x^2}}+\frac {9 x}{4 a \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {9 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{8 a^{2/3} b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {3\ 3^{3/4} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{2 \sqrt {2} a^{2/3} b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}} \]

output

3/2*x/(-b*x^2+a)^(4/3)+9/4*x/a/(-b*x^2+a)^(1/3)+9/4*x/a/(-(-b*x^2+a)^(1/3) 
+a^(1/3)*(1-3^(1/2)))-3/4*(a^(1/3)-(-b*x^2+a)^(1/3))*EllipticF((-(-b*x^2+a 
)^(1/3)+a^(1/3)*(1+3^(1/2)))/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2))),2*I-I 
*3^(1/2))*((a^(2/3)+a^(1/3)*(-b*x^2+a)^(1/3)+(-b*x^2+a)^(2/3))/(-(-b*x^2+a 
)^(1/3)+a^(1/3)*(1-3^(1/2)))^2)^(1/2)*3^(3/4)/a^(2/3)/b/x*2^(1/2)/(-a^(1/3 
)*(a^(1/3)-(-b*x^2+a)^(1/3))/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))^2)^(1 
/2)+9/8*(a^(1/3)-(-b*x^2+a)^(1/3))*EllipticE((-(-b*x^2+a)^(1/3)+a^(1/3)*(1 
+3^(1/2)))/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2))),2*I-I*3^(1/2))*((a^(2/3 
)+a^(1/3)*(-b*x^2+a)^(1/3)+(-b*x^2+a)^(2/3))/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1 
-3^(1/2)))^2)^(1/2)*(1/2*6^(1/2)+1/2*2^(1/2))*3^(1/4)/a^(2/3)/b/x/(-a^(1/3 
)*(a^(1/3)-(-b*x^2+a)^(1/3))/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))^2)^(1 
/2)
 

3.2.38.2 Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.13 \[ \int \frac {3 a+b x^2}{\left (a-b x^2\right )^{7/3}} \, dx=\frac {15 a x-9 b x^3-3 x \left (a-b x^2\right ) \sqrt [3]{1-\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\frac {b x^2}{a}\right )}{4 a \left (a-b x^2\right )^{4/3}} \]

input

Integrate[(3*a + b*x^2)/(a - b*x^2)^(7/3),x]
 

output

(15*a*x - 9*b*x^3 - 3*x*(a - b*x^2)*(1 - (b*x^2)/a)^(1/3)*Hypergeometric2F 
1[1/3, 1/2, 3/2, (b*x^2)/a])/(4*a*(a - b*x^2)^(4/3))
 

3.2.38.3 Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 641, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {298, 215, 233, 833, 760, 2418}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {3 a+b x^2}{\left (a-b x^2\right )^{7/3}} \, dx\)

\(\Big \downarrow \) 298

\(\displaystyle \frac {3}{2} \int \frac {1}{\left (a-b x^2\right )^{4/3}}dx+\frac {3 x}{2 \left (a-b x^2\right )^{4/3}}\)

\(\Big \downarrow \) 215

\(\displaystyle \frac {3}{2} \left (\frac {3 x}{2 a \sqrt [3]{a-b x^2}}-\frac {\int \frac {1}{\sqrt [3]{a-b x^2}}dx}{2 a}\right )+\frac {3 x}{2 \left (a-b x^2\right )^{4/3}}\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {3}{2} \left (\frac {3 \sqrt {-b x^2} \int \frac {\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}}{4 a b x}+\frac {3 x}{2 a \sqrt [3]{a-b x^2}}\right )+\frac {3 x}{2 \left (a-b x^2\right )^{4/3}}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {3}{2} \left (\frac {3 \sqrt {-b x^2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}\right )}{4 a b x}+\frac {3 x}{2 a \sqrt [3]{a-b x^2}}\right )+\frac {3 x}{2 \left (a-b x^2\right )^{4/3}}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {3}{2} \left (\frac {3 \sqrt {-b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}\right )}{4 a b x}+\frac {3 x}{2 a \sqrt [3]{a-b x^2}}\right )+\frac {3 x}{2 \left (a-b x^2\right )^{4/3}}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {3}{2} \left (\frac {3 \sqrt {-b x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {2 \sqrt {-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )}{4 a b x}+\frac {3 x}{2 a \sqrt [3]{a-b x^2}}\right )+\frac {3 x}{2 \left (a-b x^2\right )^{4/3}}\)

input

Int[(3*a + b*x^2)/(a - b*x^2)^(7/3),x]
 

output

(3*x)/(2*(a - b*x^2)^(4/3)) + (3*((3*x)/(2*a*(a - b*x^2)^(1/3)) + (3*Sqrt[ 
-(b*x^2)]*((-2*Sqrt[-(b*x^2)])/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3)) 
 + (3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[( 
a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^ 
(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a 
 - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt 
[3]])/(Sqrt[-(b*x^2)]*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - 
 Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) - (2*Sqrt[2 - Sqrt[3]]*(1 + Sq 
rt[3])*a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - 
b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/ 
3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - 
Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(3^(1/4)*Sqrt[-(b 
*x^2)]*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/ 
3) - (a - b*x^2)^(1/3))^2)])))/(4*a*b*x)))/2
 

3.2.38.3.1 Defintions of rubi rules used

rule 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

rule 233

Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[3*(Sqrt[b*x^2]/(2*b*x)) 
   Subst[Int[x/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b 
}, x]
 

rule 298

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( 
b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 
2*p + 3))/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, 
 c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
 

rule 760

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- 
s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) 
*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x 
] && NegQ[a]
 

rule 833

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] 
], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 + Sqrt[3]))*(s/r)   Int[1/Sqrt[a + b*x 
^3], x], x] + Simp[1/r   Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x 
]] /; FreeQ[{a, b}, x] && NegQ[a]
 

rule 2418

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 + Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 + Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x] + S 
imp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 - Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - S 
qrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[ 
3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && 
 EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
 

3.2.38.4 Maple [F]

\[\int \frac {b \,x^{2}+3 a}{\left (-b \,x^{2}+a \right )^{\frac {7}{3}}}d x\]

input

int((b*x^2+3*a)/(-b*x^2+a)^(7/3),x)
 

output

int((b*x^2+3*a)/(-b*x^2+a)^(7/3),x)
 

3.2.38.5 Fricas [F]

\[ \int \frac {3 a+b x^2}{\left (a-b x^2\right )^{7/3}} \, dx=\int { \frac {b x^{2} + 3 \, a}{{\left (-b x^{2} + a\right )}^{\frac {7}{3}}} \,d x } \]

input

integrate((b*x^2+3*a)/(-b*x^2+a)^(7/3),x, algorithm="fricas")
 

output

integral(-(b*x^2 + 3*a)*(-b*x^2 + a)^(2/3)/(b^3*x^6 - 3*a*b^2*x^4 + 3*a^2* 
b*x^2 - a^3), x)
 

3.2.38.6 Sympy [A] (verification not implemented)

Time = 4.64 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.10 \[ \int \frac {3 a+b x^2}{\left (a-b x^2\right )^{7/3}} \, dx=\frac {3 x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{3} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{a^{\frac {4}{3}}} + \frac {b x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {7}{3} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{3 a^{\frac {7}{3}}} \]

input

integrate((b*x**2+3*a)/(-b*x**2+a)**(7/3),x)
 

output

3*x*hyper((1/2, 7/3), (3/2,), b*x**2*exp_polar(2*I*pi)/a)/a**(4/3) + b*x** 
3*hyper((3/2, 7/3), (5/2,), b*x**2*exp_polar(2*I*pi)/a)/(3*a**(7/3))
 

3.2.38.7 Maxima [F]

\[ \int \frac {3 a+b x^2}{\left (a-b x^2\right )^{7/3}} \, dx=\int { \frac {b x^{2} + 3 \, a}{{\left (-b x^{2} + a\right )}^{\frac {7}{3}}} \,d x } \]

input

integrate((b*x^2+3*a)/(-b*x^2+a)^(7/3),x, algorithm="maxima")
 

output

integrate((b*x^2 + 3*a)/(-b*x^2 + a)^(7/3), x)
 

3.2.38.8 Giac [F]

\[ \int \frac {3 a+b x^2}{\left (a-b x^2\right )^{7/3}} \, dx=\int { \frac {b x^{2} + 3 \, a}{{\left (-b x^{2} + a\right )}^{\frac {7}{3}}} \,d x } \]

input

integrate((b*x^2+3*a)/(-b*x^2+a)^(7/3),x, algorithm="giac")
 

output

integrate((b*x^2 + 3*a)/(-b*x^2 + a)^(7/3), x)
 

3.2.38.9 Mupad [F(-1)]

Timed out. \[ \int \frac {3 a+b x^2}{\left (a-b x^2\right )^{7/3}} \, dx=\int \frac {b\,x^2+3\,a}{{\left (a-b\,x^2\right )}^{7/3}} \,d x \]

input

int((3*a + b*x^2)/(a - b*x^2)^(7/3),x)
 

output

int((3*a + b*x^2)/(a - b*x^2)^(7/3), x)