\(\int \frac {1}{\sqrt [4]{a+b x} (c+d x)^{3/4}} \, dx\) [603]

Optimal result

Integrand size = 19, antiderivative size = 85 \[ \int \frac {1}{\sqrt [4]{a+b x} (c+d x)^{3/4}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{b} d^{3/4}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{b} d^{3/4}} \] Output:

-2*arctan(d^(1/4)*(b*x+a)^(1/4)/b^(1/4)/(d*x+c)^(1/4))/b^(1/4)/d^(3/4)+2*a 
rctanh(d^(1/4)*(b*x+a)^(1/4)/b^(1/4)/(d*x+c)^(1/4))/b^(1/4)/d^(3/4)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\sqrt [4]{a+b x} (c+d x)^{3/4}} \, dx=\frac {2 \left (\arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )\right )}{\sqrt [4]{b} d^{3/4}} \] Input:

Integrate[1/((a + b*x)^(1/4)*(c + d*x)^(3/4)),x]
 

Output:

(2*(ArcTan[(b^(1/4)*(c + d*x)^(1/4))/(d^(1/4)*(a + b*x)^(1/4))] + ArcTanh[ 
(b^(1/4)*(c + d*x)^(1/4))/(d^(1/4)*(a + b*x)^(1/4))]))/(b^(1/4)*d^(3/4))
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.40, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {73, 854, 27, 827, 218, 221}

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.

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 854

\(\displaystyle \frac {4 \int \frac {b \sqrt {a+b x}}{b-d (a+b x)}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle 4 \int \frac {\sqrt {a+b x}}{b-d (a+b x)}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\)

\(\Big \downarrow \) 827

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

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

Input:

Int[1/((a + b*x)^(1/4)*(c + d*x)^(3/4)),x]
 

Output:

4*(-1/2*ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c - (a*d)/b + (d*(a + b 
*x))/b)^(1/4))]/(b^(1/4)*d^(3/4)) + ArcTanh[(d^(1/4)*(a + b*x)^(1/4))/(b^( 
1/4)*(c - (a*d)/b + (d*(a + b*x))/b)^(1/4))]/(2*b^(1/4)*d^(3/4)))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 221

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

rule 827

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 
 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b)   Int[1/(r + s*x^2), x], 
x] - Simp[s/(2*b)   Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ 
[a/b, 0]
 

rule 854

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 
 1)/n)   Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n 
)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 
2^(-1)] && IntegersQ[m, p + (m + 1)/n]
 

Maple [F]

\[\int \frac {1}{\left (b x +a \right )^{\frac {1}{4}} \left (x d +c \right )^{\frac {3}{4}}}d x\]

Input:

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

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.61 \[ \int \frac {1}{\sqrt [4]{a+b x} (c+d x)^{3/4}} \, dx=\left (\frac {1}{b d^{3}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b d x + a d\right )} \left (\frac {1}{b d^{3}}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{b x + a}\right ) - \left (\frac {1}{b d^{3}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (b d x + a d\right )} \left (\frac {1}{b d^{3}}\right )^{\frac {1}{4}} - {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{b x + a}\right ) + i \, \left (\frac {1}{b d^{3}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (i \, b d x + i \, a d\right )} \left (\frac {1}{b d^{3}}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{b x + a}\right ) - i \, \left (\frac {1}{b d^{3}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (-i \, b d x - i \, a d\right )} \left (\frac {1}{b d^{3}}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{b x + a}\right ) \] Input:

integrate(1/(b*x+a)^(1/4)/(d*x+c)^(3/4),x, algorithm="fricas")
 

Output:

(1/(b*d^3))^(1/4)*log(((b*d*x + a*d)*(1/(b*d^3))^(1/4) + (b*x + a)^(3/4)*( 
d*x + c)^(1/4))/(b*x + a)) - (1/(b*d^3))^(1/4)*log(-((b*d*x + a*d)*(1/(b*d 
^3))^(1/4) - (b*x + a)^(3/4)*(d*x + c)^(1/4))/(b*x + a)) + I*(1/(b*d^3))^( 
1/4)*log(((I*b*d*x + I*a*d)*(1/(b*d^3))^(1/4) + (b*x + a)^(3/4)*(d*x + c)^ 
(1/4))/(b*x + a)) - I*(1/(b*d^3))^(1/4)*log(((-I*b*d*x - I*a*d)*(1/(b*d^3) 
)^(1/4) + (b*x + a)^(3/4)*(d*x + c)^(1/4))/(b*x + a))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(1/(b*x+a)^(1/4)/(d*x+c)^(3/4),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(1/(b*x+a)^(1/4)/(d*x+c)^(3/4),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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