\(\int \frac {x^3 (c+d x)}{(a+b x^2)^{4/3}} \, dx\) [1627]

Optimal result

Integrand size = 20, antiderivative size = 603 \[ \int \frac {x^3 (c+d x)}{\left (a+b x^2\right )^{4/3}} \, dx=\frac {3 a (c+d x)}{2 b^2 \sqrt [3]{a+b x^2}}+\frac {3 c \left (a+b x^2\right )^{2/3}}{4 b^2}+\frac {3 d x \left (a+b x^2\right )^{2/3}}{7 b^2}+\frac {81 a d x}{14 b^2 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}-\frac {81 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{4/3} d \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )|-7+4 \sqrt {3}\right )}{28 b^3 x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac {27\ 3^{3/4} a^{4/3} d \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right ),-7+4 \sqrt {3}\right )}{7 \sqrt {2} b^3 x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}} \] Output:

3/2*a*(d*x+c)/b^2/(b*x^2+a)^(1/3)+3/4*c*(b*x^2+a)^(2/3)/b^2+3/7*d*x*(b*x^2 
+a)^(2/3)/b^2+81/14*a*d*x/b^2/((1-3^(1/2))*a^(1/3)-(b*x^2+a)^(1/3))-81/28* 
3^(1/4)*(1/2*6^(1/2)+1/2*2^(1/2))*a^(4/3)*d*(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))/b^3/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)+27/14*3^(3/4 
)*a^(4/3)*d*(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)*EllipticF((( 
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)/b^3/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)
 

Mathematica [C] (verified)

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

Time = 10.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.14 \[ \int \frac {x^3 (c+d x)}{\left (a+b x^2\right )^{4/3}} \, dx=\frac {3 b x^2 (7 c+4 d x)+9 a (7 c+6 d x)-54 a d x \sqrt [3]{1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-\frac {b x^2}{a}\right )}{28 b^2 \sqrt [3]{a+b x^2}} \] Input:

Integrate[(x^3*(c + d*x))/(a + b*x^2)^(4/3),x]
 

Output:

(3*b*x^2*(7*c + 4*d*x) + 9*a*(7*c + 6*d*x) - 54*a*d*x*(1 + (b*x^2)/a)^(1/3 
)*Hypergeometric2F1[1/3, 1/2, 3/2, -((b*x^2)/a)])/(28*b^2*(a + b*x^2)^(1/3 
))
 

Rubi [A] (warning: unable to verify)

Time = 0.65 (sec) , antiderivative size = 668, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {542, 243, 53, 252, 262, 233, 833, 760, 2009, 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.

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

\(\Big \downarrow \) 542

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 53

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

\(\Big \downarrow \) 252

\(\displaystyle \frac {1}{2} c \int \left (\frac {1}{b \sqrt [3]{b x^2+a}}-\frac {a}{b \left (b x^2+a\right )^{4/3}}\right )dx^2+d \left (\frac {9 \int \frac {x^2}{\sqrt [3]{b x^2+a}}dx}{2 b}-\frac {3 x^3}{2 b \sqrt [3]{a+b x^2}}\right )\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {1}{2} c \int \left (\frac {1}{b \sqrt [3]{b x^2+a}}-\frac {a}{b \left (b x^2+a\right )^{4/3}}\right )dx^2+d \left (\frac {9 \left (\frac {3 x \left (a+b x^2\right )^{2/3}}{7 b}-\frac {3 a \int \frac {1}{\sqrt [3]{b x^2+a}}dx}{7 b}\right )}{2 b}-\frac {3 x^3}{2 b \sqrt [3]{a+b x^2}}\right )\)

\(\Big \downarrow \) 233

\(\displaystyle d \left (\frac {9 \left (\frac {3 x \left (a+b x^2\right )^{2/3}}{7 b}-\frac {9 a \sqrt {b x^2} \int \frac {\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}}{14 b^2 x}\right )}{2 b}-\frac {3 x^3}{2 b \sqrt [3]{a+b x^2}}\right )+\frac {1}{2} c \int \left (\frac {1}{b \sqrt [3]{b x^2+a}}-\frac {a}{b \left (b x^2+a\right )^{4/3}}\right )dx^2\)

\(\Big \downarrow \) 833

\(\displaystyle d \left (\frac {9 \left (\frac {3 x \left (a+b x^2\right )^{2/3}}{7 b}-\frac {9 a \sqrt {b x^2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}\right )}{14 b^2 x}\right )}{2 b}-\frac {3 x^3}{2 b \sqrt [3]{a+b x^2}}\right )+\frac {1}{2} c \int \left (\frac {1}{b \sqrt [3]{b x^2+a}}-\frac {a}{b \left (b x^2+a\right )^{4/3}}\right )dx^2\)

\(\Big \downarrow \) 760

\(\displaystyle d \left (\frac {9 \left (\frac {3 x \left (a+b x^2\right )^{2/3}}{7 b}-\frac {9 a \sqrt {b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}\right )}{14 b^2 x}\right )}{2 b}-\frac {3 x^3}{2 b \sqrt [3]{a+b x^2}}\right )+\frac {1}{2} c \int \left (\frac {1}{b \sqrt [3]{b x^2+a}}-\frac {a}{b \left (b x^2+a\right )^{4/3}}\right )dx^2\)

\(\Big \downarrow \) 2009

\(\displaystyle d \left (\frac {9 \left (\frac {3 x \left (a+b x^2\right )^{2/3}}{7 b}-\frac {9 a \sqrt {b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}\right )}{14 b^2 x}\right )}{2 b}-\frac {3 x^3}{2 b \sqrt [3]{a+b x^2}}\right )+\frac {1}{2} c \left (\frac {3 a}{b^2 \sqrt [3]{a+b x^2}}+\frac {3 \left (a+b x^2\right )^{2/3}}{2 b^2}\right )\)

\(\Big \downarrow \) 2418

\(\displaystyle d \left (\frac {9 \left (\frac {3 x \left (a+b x^2\right )^{2/3}}{7 b}-\frac {9 a \sqrt {b x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac {2 \sqrt {b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )}{14 b^2 x}\right )}{2 b}-\frac {3 x^3}{2 b \sqrt [3]{a+b x^2}}\right )+\frac {1}{2} c \left (\frac {3 a}{b^2 \sqrt [3]{a+b x^2}}+\frac {3 \left (a+b x^2\right )^{2/3}}{2 b^2}\right )\)

Input:

Int[(x^3*(c + d*x))/(a + b*x^2)^(4/3),x]
 

Output:

(c*((3*a)/(b^2*(a + b*x^2)^(1/3)) + (3*(a + b*x^2)^(2/3))/(2*b^2)))/2 + d* 
((-3*x^3)/(2*b*(a + b*x^2)^(1/3)) + (9*((3*x*(a + b*x^2)^(2/3))/(7*b) - (9 
*a*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*Sq 
rt[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 + Sqr 
t[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 - S 
qrt[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)])))/(14*b^2*x)))/(2*b))
 

Defintions of rubi rules used

rule 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

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 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 252

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x 
)^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* 
(p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c 
}, x] && LtQ[p, -1] && GtQ[m, 1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi 
alQ[a, b, c, 2, m, p, x]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 542

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 Simp[c   Int[x^m*(a + b*x^2)^p, x], x] + Simp[d   Int[x^(m + 1)*(a + b*x^2 
)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && IntegerQ[m] &&  !IntegerQ[2*p]
 

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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]
 

Maple [F]

\[\int \frac {x^{3} \left (d x +c \right )}{\left (b \,x^{2}+a \right )^{\frac {4}{3}}}d x\]

Input:

int(x^3*(d*x+c)/(b*x^2+a)^(4/3),x)
 

Output:

int(x^3*(d*x+c)/(b*x^2+a)^(4/3),x)
 

Fricas [F]

\[ \int \frac {x^3 (c+d x)}{\left (a+b x^2\right )^{4/3}} \, dx=\int { \frac {{\left (d x + c\right )} x^{3}}{{\left (b x^{2} + a\right )}^{\frac {4}{3}}} \,d x } \] Input:

integrate(x^3*(d*x+c)/(b*x^2+a)^(4/3),x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 2.68 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.13 \[ \int \frac {x^3 (c+d x)}{\left (a+b x^2\right )^{4/3}} \, dx=c \left (\begin {cases} \frac {9 a}{4 b^{2} \sqrt [3]{a + b x^{2}}} + \frac {3 x^{2}}{4 b \sqrt [3]{a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{\frac {4}{3}}} & \text {otherwise} \end {cases}\right ) + \frac {d x^{5} {{}_{2}F_{1}\left (\begin {matrix} \frac {4}{3}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{5 a^{\frac {4}{3}}} \] Input:

integrate(x**3*(d*x+c)/(b*x**2+a)**(4/3),x)
 

Output:

c*Piecewise((9*a/(4*b**2*(a + b*x**2)**(1/3)) + 3*x**2/(4*b*(a + b*x**2)** 
(1/3)), Ne(b, 0)), (x**4/(4*a**(4/3)), True)) + d*x**5*hyper((4/3, 5/2), ( 
7/2,), b*x**2*exp_polar(I*pi)/a)/(5*a**(4/3))
 

Maxima [F]

\[ \int \frac {x^3 (c+d x)}{\left (a+b x^2\right )^{4/3}} \, dx=\int { \frac {{\left (d x + c\right )} x^{3}}{{\left (b x^{2} + a\right )}^{\frac {4}{3}}} \,d x } \] Input:

integrate(x^3*(d*x+c)/(b*x^2+a)^(4/3),x, algorithm="maxima")
 

Output:

3/4*c*((b*x^2 + a)^(2/3)/b^2 + 2*a/((b*x^2 + a)^(1/3)*b^2)) + d*integrate( 
x^4/(b*x^2 + a)^(4/3), x)
 

Giac [F]

\[ \int \frac {x^3 (c+d x)}{\left (a+b x^2\right )^{4/3}} \, dx=\int { \frac {{\left (d x + c\right )} x^{3}}{{\left (b x^{2} + a\right )}^{\frac {4}{3}}} \,d x } \] Input:

integrate(x^3*(d*x+c)/(b*x^2+a)^(4/3),x, algorithm="giac")
 

Output:

integrate((d*x + c)*x^3/(b*x^2 + a)^(4/3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 (c+d x)}{\left (a+b x^2\right )^{4/3}} \, dx=\int \frac {x^3\,\left (c+d\,x\right )}{{\left (b\,x^2+a\right )}^{4/3}} \,d x \] Input:

int((x^3*(c + d*x))/(a + b*x^2)^(4/3),x)
                                                                                    
                                                                                    
 

Output:

int((x^3*(c + d*x))/(a + b*x^2)^(4/3), x)
 

Reduce [F]

\[ \int \frac {x^3 (c+d x)}{\left (a+b x^2\right )^{4/3}} \, dx=\left (\int \frac {x^{4}}{\left (b \,x^{2}+a \right )^{\frac {1}{3}} a +\left (b \,x^{2}+a \right )^{\frac {1}{3}} b \,x^{2}}d x \right ) d +\left (\int \frac {x^{3}}{\left (b \,x^{2}+a \right )^{\frac {1}{3}} a +\left (b \,x^{2}+a \right )^{\frac {1}{3}} b \,x^{2}}d x \right ) c \] Input:

int(x^3*(d*x+c)/(b*x^2+a)^(4/3),x)
 

Output:

int(x**4/((a + b*x**2)**(1/3)*a + (a + b*x**2)**(1/3)*b*x**2),x)*d + int(x 
**3/((a + b*x**2)**(1/3)*a + (a + b*x**2)**(1/3)*b*x**2),x)*c