3.18.6 \(\int \frac {(a+b x)^{3/4}}{(c+d x)^{3/4}} \, dx\) [1706]

3.18.6.1 Optimal result

Integrand size = 19, antiderivative size = 127 \[ \int \frac {(a+b x)^{3/4}}{(c+d x)^{3/4}} \, dx=\frac {(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}+\frac {3 (b c-a d) \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 \sqrt [4]{b} d^{7/4}}-\frac {3 (b c-a d) \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 \sqrt [4]{b} d^{7/4}} \]

output

(b*x+a)^(3/4)*(d*x+c)^(1/4)/d+3/2*(-a*d+b*c)*arctan(d^(1/4)*(b*x+a)^(1/4)/ 
b^(1/4)/(d*x+c)^(1/4))/b^(1/4)/d^(7/4)-3/2*(-a*d+b*c)*arctanh(d^(1/4)*(b*x 
+a)^(1/4)/b^(1/4)/(d*x+c)^(1/4))/b^(1/4)/d^(7/4)
 

3.18.6.2 Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^{3/4}}{(c+d x)^{3/4}} \, dx=\frac {(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}-\frac {3 (b c-a d) \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )}{2 \sqrt [4]{b} d^{7/4}}-\frac {3 (b c-a d) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )}{2 \sqrt [4]{b} d^{7/4}} \]

input

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

output

((a + b*x)^(3/4)*(c + d*x)^(1/4))/d - (3*(b*c - a*d)*ArcTan[(b^(1/4)*(c + 
d*x)^(1/4))/(d^(1/4)*(a + b*x)^(1/4))])/(2*b^(1/4)*d^(7/4)) - (3*(b*c - a* 
d)*ArcTanh[(b^(1/4)*(c + d*x)^(1/4))/(d^(1/4)*(a + b*x)^(1/4))])/(2*b^(1/4 
)*d^(7/4))
 

3.18.6.3 Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.20, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {60, 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.

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 854

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 827

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

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

input

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

output

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

3.18.6.3.1 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 60

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

3.18.6.4 Maple [F]

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

input

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

output

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

3.18.6.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.24 (sec) , antiderivative size = 691, normalized size of antiderivative = 5.44 \[ \int \frac {(a+b x)^{3/4}}{(c+d x)^{3/4}} \, dx =\text {Too large to display} \]

input

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

output

-1/4*(3*d*((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + 
a^4*d^4)/(b*d^7))^(1/4)*log(-3*((b*c - a*d)*(b*x + a)^(3/4)*(d*x + c)^(1/4 
) + (b*d^2*x + a*d^2)*((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^ 
3*b*c*d^3 + a^4*d^4)/(b*d^7))^(1/4))/(b*x + a)) - 3*d*((b^4*c^4 - 4*a*b^3* 
c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(b*d^7))^(1/4)*log(-3 
*((b*c - a*d)*(b*x + a)^(3/4)*(d*x + c)^(1/4) - (b*d^2*x + a*d^2)*((b^4*c^ 
4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(b*d^7))^ 
(1/4))/(b*x + a)) + 3*I*d*((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 
4*a^3*b*c*d^3 + a^4*d^4)/(b*d^7))^(1/4)*log(-3*((b*c - a*d)*(b*x + a)^(3/4 
)*(d*x + c)^(1/4) + (I*b*d^2*x + I*a*d^2)*((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^ 
2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(b*d^7))^(1/4))/(b*x + a)) - 3*I* 
d*((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4) 
/(b*d^7))^(1/4)*log(-3*((b*c - a*d)*(b*x + a)^(3/4)*(d*x + c)^(1/4) + (-I* 
b*d^2*x - I*a*d^2)*((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b 
*c*d^3 + a^4*d^4)/(b*d^7))^(1/4))/(b*x + a)) - 4*(b*x + a)^(3/4)*(d*x + c) 
^(1/4))/d
 

3.18.6.6 Sympy [F]

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

input

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

output

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

3.18.6.7 Maxima [F]

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

input

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

output

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

3.18.6.8 Giac [F]

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

input

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

output

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

3.18.6.9 Mupad [F(-1)]

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

input

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

output

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

3.18.6.10 Reduce [F]

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

input

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

output

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