\(\int \frac {1}{(a+b x)^{5/2} \sqrt [3]{c+d x}} \, dx\) [425]

Optimal result

Integrand size = 19, antiderivative size = 842 \[ \int \frac {1}{(a+b x)^{5/2} \sqrt [3]{c+d x}} \, dx=-\frac {2 (c+d x)^{2/3}}{3 (b c-a d) (a+b x)^{3/2}}+\frac {10 d (c+d x)^{2/3}}{9 (b c-a d)^2 \sqrt {a+b x}}+\frac {10 d^2 \sqrt {a+b x}}{9 b^{2/3} (b c-a d)^2 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}-\frac {5 \sqrt {2+\sqrt {3}} d \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt {3}\right )}{3\ 3^{3/4} b^{2/3} (b c-a d)^{5/3} \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac {10 \sqrt {2} d \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right ),-7+4 \sqrt {3}\right )}{9 \sqrt [4]{3} b^{2/3} (b c-a d)^{5/3} \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}} \] Output:

-2/3*(d*x+c)^(2/3)/(-a*d+b*c)/(b*x+a)^(3/2)+10/9*d*(d*x+c)^(2/3)/(-a*d+b*c 
)^2/(b*x+a)^(1/2)+10/9*d^2*(b*x+a)^(1/2)/b^(2/3)/(-a*d+b*c)^2/((1-3^(1/2)) 
*(-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3))-5/9*(1/2*6^(1/2)+1/2*2^(1/2))*d*( 
(-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3))*(((-a*d+b*c)^(2/3)+b^(1/3)*(-a*d+b 
*c)^(1/3)*(d*x+c)^(1/3)+b^(2/3)*(d*x+c)^(2/3))/((1-3^(1/2))*(-a*d+b*c)^(1/ 
3)-b^(1/3)*(d*x+c)^(1/3))^2)^(1/2)*EllipticE(((1+3^(1/2))*(-a*d+b*c)^(1/3) 
-b^(1/3)*(d*x+c)^(1/3))/((1-3^(1/2))*(-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3 
)),2*I-I*3^(1/2))*3^(1/4)/b^(2/3)/(-a*d+b*c)^(5/3)/(b*x+a)^(1/2)/(-(-a*d+b 
*c)^(1/3)*((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3))/((1-3^(1/2))*(-a*d+b*c) 
^(1/3)-b^(1/3)*(d*x+c)^(1/3))^2)^(1/2)+10/27*2^(1/2)*d*((-a*d+b*c)^(1/3)-b 
^(1/3)*(d*x+c)^(1/3))*(((-a*d+b*c)^(2/3)+b^(1/3)*(-a*d+b*c)^(1/3)*(d*x+c)^ 
(1/3)+b^(2/3)*(d*x+c)^(2/3))/((1-3^(1/2))*(-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c) 
^(1/3))^2)^(1/2)*EllipticF(((1+3^(1/2))*(-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^( 
1/3))/((1-3^(1/2))*(-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3)),2*I-I*3^(1/2))* 
3^(3/4)/b^(2/3)/(-a*d+b*c)^(5/3)/(b*x+a)^(1/2)/(-(-a*d+b*c)^(1/3)*((-a*d+b 
*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3))/((1-3^(1/2))*(-a*d+b*c)^(1/3)-b^(1/3)*(d* 
x+c)^(1/3))^2)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

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

Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.09 \[ \int \frac {1}{(a+b x)^{5/2} \sqrt [3]{c+d x}} \, dx=-\frac {2 \sqrt [3]{\frac {b (c+d x)}{b c-a d}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{3},-\frac {1}{2},\frac {d (a+b x)}{-b c+a d}\right )}{3 b (a+b x)^{3/2} \sqrt [3]{c+d x}} \] Input:

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

Output:

(-2*((b*(c + d*x))/(b*c - a*d))^(1/3)*Hypergeometric2F1[-3/2, 1/3, -1/2, ( 
d*(a + b*x))/(-(b*c) + a*d)])/(3*b*(a + b*x)^(3/2)*(c + d*x)^(1/3))
 

Rubi [A] (warning: unable to verify)

Time = 0.72 (sec) , antiderivative size = 912, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {61, 61, 73, 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.

\(\displaystyle \int \frac {1}{(a+b x)^{5/2} \sqrt [3]{c+d x}} \, dx\)

\(\Big \downarrow \) 61

\(\displaystyle -\frac {5 d \int \frac {1}{(a+b x)^{3/2} \sqrt [3]{c+d x}}dx}{9 (b c-a d)}-\frac {2 (c+d x)^{2/3}}{3 (a+b x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 61

\(\displaystyle -\frac {5 d \left (\frac {d \int \frac {1}{\sqrt {a+b x} \sqrt [3]{c+d x}}dx}{3 (b c-a d)}-\frac {2 (c+d x)^{2/3}}{\sqrt {a+b x} (b c-a d)}\right )}{9 (b c-a d)}-\frac {2 (c+d x)^{2/3}}{3 (a+b x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {5 d \left (\frac {\int \frac {\sqrt [3]{c+d x}}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [3]{c+d x}}{b c-a d}-\frac {2 (c+d x)^{2/3}}{\sqrt {a+b x} (b c-a d)}\right )}{9 (b c-a d)}-\frac {2 (c+d x)^{2/3}}{3 (a+b x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 833

\(\displaystyle -\frac {5 d \left (\frac {\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d} \int \frac {1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [3]{c+d x}}{\sqrt [3]{b}}-\frac {\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [3]{c+d x}}{\sqrt [3]{b}}}{b c-a d}-\frac {2 (c+d x)^{2/3}}{\sqrt {a+b x} (b c-a d)}\right )}{9 (b c-a d)}-\frac {2 (c+d x)^{2/3}}{3 (a+b x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 760

\(\displaystyle -\frac {5 d \left (\frac {-\frac {\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [3]{c+d x}}{\sqrt [3]{b}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {-\frac {\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}}{b c-a d}-\frac {2 (c+d x)^{2/3}}{\sqrt {a+b x} (b c-a d)}\right )}{9 (b c-a d)}-\frac {2 (c+d x)^{2/3}}{3 (a+b x)^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 2418

\(\displaystyle -\frac {5 d \left (\frac {-\frac {\frac {2 d \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}{\sqrt [3]{b} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}-\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {-\frac {\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}}{\sqrt [3]{b}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {-\frac {\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}}{b c-a d}-\frac {2 (c+d x)^{2/3}}{(b c-a d) \sqrt {a+b x}}\right )}{9 (b c-a d)}-\frac {2 (c+d x)^{2/3}}{3 (b c-a d) (a+b x)^{3/2}}\)

Input:

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

Output:

(-2*(c + d*x)^(2/3))/(3*(b*c - a*d)*(a + b*x)^(3/2)) - (5*d*((-2*(c + d*x) 
^(2/3))/((b*c - a*d)*Sqrt[a + b*x]) + (-(((2*d*Sqrt[a - (b*c)/d + (b*(c + 
d*x))/d])/(b^(1/3)*((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1 
/3))) - (3^(1/4)*Sqrt[2 + Sqrt[3]]*(b*c - a*d)^(1/3)*((b*c - a*d)^(1/3) - 
b^(1/3)*(c + d*x)^(1/3))*Sqrt[((b*c - a*d)^(2/3) + b^(1/3)*(b*c - a*d)^(1/ 
3)*(c + d*x)^(1/3) + b^(2/3)*(c + d*x)^(2/3))/((1 - Sqrt[3])*(b*c - a*d)^( 
1/3) - b^(1/3)*(c + d*x)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*(b*c - 
a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b 
^(1/3)*(c + d*x)^(1/3))], -7 + 4*Sqrt[3]])/(b^(1/3)*Sqrt[-(((b*c - a*d)^(1 
/3)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)))/((1 - Sqrt[3])*(b*c - a 
*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))^2)]*Sqrt[a - (b*c)/d + (b*(c + d*x))/ 
d]))/b^(1/3)) - (2*Sqrt[2 - Sqrt[3]]*(1 + Sqrt[3])*(b*c - a*d)^(1/3)*((b*c 
 - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))*Sqrt[((b*c - a*d)^(2/3) + b^(1/3) 
*(b*c - a*d)^(1/3)*(c + d*x)^(1/3) + b^(2/3)*(c + d*x)^(2/3))/((1 - Sqrt[3 
])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))^2]*EllipticF[ArcSin[((1 + 
Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))/((1 - Sqrt[3])*(b*c 
- a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))], -7 + 4*Sqrt[3]])/(3^(1/4)*b^(2/3 
)*Sqrt[-(((b*c - a*d)^(1/3)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))) 
/((1 - Sqrt[3])*(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))^2)]*Sqrt[a - 
(b*c)/d + (b*(c + d*x))/d]))/(b*c - a*d)))/(9*(b*c - a*d))
 

Defintions of rubi rules used

rule 61

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0 
] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d 
, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

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]
 

Maple [F]

\[\int \frac {1}{\left (b x +a \right )^{\frac {5}{2}} \left (x d +c \right )^{\frac {1}{3}}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {1}{(a+b x)^{5/2} \sqrt [3]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{2}} {\left (d x + c\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate(1/(b*x+a)^(5/2)/(d*x+c)^(1/3),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {1}{(a+b x)^{5/2} \sqrt [3]{c+d x}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {5}{2}} \sqrt [3]{c + d x}}\, dx \] Input:

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

Output:

Integral(1/((a + b*x)**(5/2)*(c + d*x)**(1/3)), x)
 

Maxima [F]

\[ \int \frac {1}{(a+b x)^{5/2} \sqrt [3]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{2}} {\left (d x + c\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate(1/(b*x+a)^(5/2)/(d*x+c)^(1/3),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {1}{(a+b x)^{5/2} \sqrt [3]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{2}} {\left (d x + c\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate(1/(b*x+a)^(5/2)/(d*x+c)^(1/3),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x)^{5/2} \sqrt [3]{c+d x}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{1/3}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{(a+b x)^{5/2} \sqrt [3]{c+d x}} \, dx=\int \frac {\sqrt {b x +a}}{\left (d x +c \right )^{\frac {1}{3}} a^{3}+3 \left (d x +c \right )^{\frac {1}{3}} a^{2} b x +3 \left (d x +c \right )^{\frac {1}{3}} a \,b^{2} x^{2}+\left (d x +c \right )^{\frac {1}{3}} b^{3} x^{3}}d x \] Input:

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

Output:

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