3.16.73 \(\int \frac {1}{(a+b x)^{3/2} (c+d x)^{2/3}} \, dx\) [1573]

3.16.73.1 Optimal result

Integrand size = 19, antiderivative size = 383 \[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{2/3}} \, dx=-\frac {2 \sqrt [3]{c+d x}}{(b c-a d) \sqrt {a+b x}}+\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{b} (b c-a d) \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}} \]

output

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

3.16.73.2 Mathematica [C] (verified)

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

Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.19 \[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{2/3}} \, dx=-\frac {2 \left (\frac {b (c+d x)}{b c-a d}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {2}{3},\frac {1}{2},\frac {d (a+b x)}{-b c+a d}\right )}{b \sqrt {a+b x} (c+d x)^{2/3}} \]

input

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

output

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

3.16.73.3 Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {61, 73, 760}

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.

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

\(\Big \downarrow \) 61

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 760

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

input

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

output

(-2*(c + d*x)^(1/3))/((b*c - a*d)*Sqrt[a + b*x]) + (2*Sqrt[2 - Sqrt[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 - Sqr 
t[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^( 
1/3)*(b*c - a*d)*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])
 

3.16.73.3.1 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]
 

3.16.73.4 Maple [F]

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

input

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

output

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

3.16.73.5 Fricas [F]

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

input

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

output

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

3.16.73.6 Sympy [F]

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

input

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

output

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

3.16.73.7 Maxima [F]

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

input

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

output

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

3.16.73.8 Giac [F]

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

input

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

output

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

3.16.73.9 Mupad [F(-1)]

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

input

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

output

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