Integrand size = 17, antiderivative size = 566 \[ \int \frac {c+d x}{\left (a+b x^2\right )^{4/3}} \, dx=-\frac {3 (a d-b c x)}{2 a b \sqrt [3]{a+b x^2}}+\frac {3 c x}{2 a \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} c \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )|-7+4 \sqrt {3}\right )}{4 a^{2/3} b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac {3^{3/4} c \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt {2} a^{2/3} b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}} \] Output:
1/2*(3*b*c*x-3*a*d)/a/b/(b*x^2+a)^(1/3)+3/2*c*x/a/((1-3^(1/2))*a^(1/3)-(b* x^2+a)^(1/3))-3/4*3^(1/4)*(1/2*6^(1/2)+1/2*2^(1/2))*c*(a^(1/3)-(b*x^2+a)^( 1/3))*((a^(2/3)+a^(1/3)*(b*x^2+a)^(1/3)+(b*x^2+a)^(2/3))/((1-3^(1/2))*a^(1 /3)-(b*x^2+a)^(1/3))^2)^(1/2)*EllipticE(((1+3^(1/2))*a^(1/3)-(b*x^2+a)^(1/ 3))/((1-3^(1/2))*a^(1/3)-(b*x^2+a)^(1/3)),2*I-I*3^(1/2))/a^(2/3)/b/x/(-a^( 1/3)*(a^(1/3)-(b*x^2+a)^(1/3))/((1-3^(1/2))*a^(1/3)-(b*x^2+a)^(1/3))^2)^(1 /2)+1/2*3^(3/4)*c*(a^(1/3)-(b*x^2+a)^(1/3))*((a^(2/3)+a^(1/3)*(b*x^2+a)^(1 /3)+(b*x^2+a)^(2/3))/((1-3^(1/2))*a^(1/3)-(b*x^2+a)^(1/3))^2)^(1/2)*Ellipt icF(((1+3^(1/2))*a^(1/3)-(b*x^2+a)^(1/3))/((1-3^(1/2))*a^(1/3)-(b*x^2+a)^( 1/3)),2*I-I*3^(1/2))*2^(1/2)/a^(2/3)/b/x/(-a^(1/3)*(a^(1/3)-(b*x^2+a)^(1/3 ))/((1-3^(1/2))*a^(1/3)-(b*x^2+a)^(1/3))^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.12 \[ \int \frac {c+d x}{\left (a+b x^2\right )^{4/3}} \, dx=\frac {-3 a d+3 b c x-b c x \sqrt [3]{1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-\frac {b x^2}{a}\right )}{2 a b \sqrt [3]{a+b x^2}} \] Input:
Integrate[(c + d*x)/(a + b*x^2)^(4/3),x]
Output:
(-3*a*d + 3*b*c*x - b*c*x*(1 + (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2 , 3/2, -((b*x^2)/a)])/(2*a*b*(a + b*x^2)^(1/3))
Time = 0.53 (sec) , antiderivative size = 609, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {454, 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.
\(\displaystyle \int \frac {c+d x}{\left (a+b x^2\right )^{4/3}} \, dx\) |
\(\Big \downarrow \) 454 |
\(\displaystyle -\frac {c \int \frac {1}{\sqrt [3]{b x^2+a}}dx}{2 a}-\frac {3 (a d-b c x)}{2 a b \sqrt [3]{a+b x^2}}\) |
\(\Big \downarrow \) 233 |
\(\displaystyle -\frac {3 c \sqrt {b x^2} \int \frac {\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}}{4 a b x}-\frac {3 (a d-b c x)}{2 a b \sqrt [3]{a+b x^2}}\) |
\(\Big \downarrow \) 833 |
\(\displaystyle -\frac {3 c \sqrt {b x^2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}\right )}{4 a b x}-\frac {3 (a d-b c x)}{2 a b \sqrt [3]{a+b x^2}}\) |
\(\Big \downarrow \) 760 |
\(\displaystyle -\frac {3 c \sqrt {b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}\right )}{4 a b x}-\frac {3 (a d-b c x)}{2 a b \sqrt [3]{a+b x^2}}\) |
\(\Big \downarrow \) 2418 |
\(\displaystyle -\frac {3 c \sqrt {b x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac {2 \sqrt {b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )}{4 a b x}-\frac {3 (a d-b c x)}{2 a b \sqrt [3]{a+b x^2}}\) |
Input:
Int[(c + d*x)/(a + b*x^2)^(4/3),x]
Output:
(-3*(a*d - b*c*x))/(2*a*b*(a + b*x^2)^(1/3)) - (3*c*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 + 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]*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 + 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]*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)
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]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*d - b*c*x)/(2*a*b*(p + 1)))*(a + b*x^2)^(p + 1), x] + Simp[c*((2*p + 3)/(2*a *(p + 1))) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] && L tQ[p, -1] && NeQ[p, -3/2]
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]
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]
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]
\[\int \frac {d x +c}{\left (b \,x^{2}+a \right )^{\frac {4}{3}}}d x\]
Input:
int((d*x+c)/(b*x^2+a)^(4/3),x)
Output:
int((d*x+c)/(b*x^2+a)^(4/3),x)
\[ \int \frac {c+d x}{\left (a+b x^2\right )^{4/3}} \, dx=\int { \frac {d x + c}{{\left (b x^{2} + a\right )}^{\frac {4}{3}}} \,d x } \] Input:
integrate((d*x+c)/(b*x^2+a)^(4/3),x, algorithm="fricas")
Output:
integral((b*x^2 + a)^(2/3)*(d*x + c)/(b^2*x^4 + 2*a*b*x^2 + a^2), x)
Time = 2.22 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.10 \[ \int \frac {c+d x}{\left (a+b x^2\right )^{4/3}} \, dx=d \left (\begin {cases} - \frac {3}{2 b \sqrt [3]{a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {4}{3}}} & \text {otherwise} \end {cases}\right ) + \frac {c x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {4}{3} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{a^{\frac {4}{3}}} \] Input:
integrate((d*x+c)/(b*x**2+a)**(4/3),x)
Output:
d*Piecewise((-3/(2*b*(a + b*x**2)**(1/3)), Ne(b, 0)), (x**2/(2*a**(4/3)), True)) + c*x*hyper((1/2, 4/3), (3/2,), b*x**2*exp_polar(I*pi)/a)/a**(4/3)
\[ \int \frac {c+d x}{\left (a+b x^2\right )^{4/3}} \, dx=\int { \frac {d x + c}{{\left (b x^{2} + a\right )}^{\frac {4}{3}}} \,d x } \] Input:
integrate((d*x+c)/(b*x^2+a)^(4/3),x, algorithm="maxima")
Output:
integrate((d*x + c)/(b*x^2 + a)^(4/3), x)
\[ \int \frac {c+d x}{\left (a+b x^2\right )^{4/3}} \, dx=\int { \frac {d x + c}{{\left (b x^{2} + a\right )}^{\frac {4}{3}}} \,d x } \] Input:
integrate((d*x+c)/(b*x^2+a)^(4/3),x, algorithm="giac")
Output:
integrate((d*x + c)/(b*x^2 + a)^(4/3), x)
Time = 8.90 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.10 \[ \int \frac {c+d x}{\left (a+b x^2\right )^{4/3}} \, dx=\frac {c\,x\,{\left (\frac {b\,x^2}{a}+1\right )}^{4/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {4}{3};\ \frac {3}{2};\ -\frac {b\,x^2}{a}\right )}{{\left (b\,x^2+a\right )}^{4/3}}-\frac {3\,d}{2\,b\,{\left (b\,x^2+a\right )}^{1/3}} \] Input:
int((c + d*x)/(a + b*x^2)^(4/3),x)
Output:
(c*x*((b*x^2)/a + 1)^(4/3)*hypergeom([1/2, 4/3], 3/2, -(b*x^2)/a))/(a + b* x^2)^(4/3) - (3*d)/(2*b*(a + b*x^2)^(1/3))
\[ \int \frac {c+d x}{\left (a+b x^2\right )^{4/3}} \, dx=\left (\int \frac {x}{\left (b \,x^{2}+a \right )^{\frac {1}{3}} a +\left (b \,x^{2}+a \right )^{\frac {1}{3}} b \,x^{2}}d x \right ) d +\left (\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{3}} a +\left (b \,x^{2}+a \right )^{\frac {1}{3}} b \,x^{2}}d x \right ) c \] Input:
int((d*x+c)/(b*x^2+a)^(4/3),x)
Output:
int(x/((a + b*x**2)**(1/3)*a + (a + b*x**2)**(1/3)*b*x**2),x)*d + int(1/(( a + b*x**2)**(1/3)*a + (a + b*x**2)**(1/3)*b*x**2),x)*c