Integrand size = 19, antiderivative size = 127 \[ \int \frac {(a+b x)^{3/4}}{(c+d x)^{3/4}} \, dx=\frac {(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}+\frac {3 (b c-a d) \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 \sqrt [4]{b} d^{7/4}}-\frac {3 (b c-a d) \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 \sqrt [4]{b} d^{7/4}} \]
(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)
Time = 0.22 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^{3/4}}{(c+d x)^{3/4}} \, dx=\frac {(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}-\frac {3 (b c-a d) \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )}{2 \sqrt [4]{b} d^{7/4}}-\frac {3 (b c-a d) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )}{2 \sqrt [4]{b} d^{7/4}} \]
((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))
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 |
\(\displaystyle \frac {(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}-\frac {3 (b c-a d) \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{2 \sqrt [4]{b} d^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{2 \sqrt [4]{b} d^{3/4}}\right )}{d}\) |
((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
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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]
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]
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]
\[\int \frac {\left (b x +a \right )^{\frac {3}{4}}}{\left (d x +c \right )^{\frac {3}{4}}}d x\]
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=-\frac {3 \, d \left (\frac {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}}\right )^{\frac {1}{4}} \log \left (-\frac {3 \, {\left ({\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} + {\left (b d^{2} x + a d^{2}\right )} \left (\frac {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}}\right )^{\frac {1}{4}}\right )}}{b x + a}\right ) - 3 \, d \left (\frac {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}}\right )^{\frac {1}{4}} \log \left (-\frac {3 \, {\left ({\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} - {\left (b d^{2} x + a d^{2}\right )} \left (\frac {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}}\right )^{\frac {1}{4}}\right )}}{b x + a}\right ) + 3 i \, d \left (\frac {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}}\right )^{\frac {1}{4}} \log \left (-\frac {3 \, {\left ({\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} + {\left (i \, b d^{2} x + i \, a d^{2}\right )} \left (\frac {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}}\right )^{\frac {1}{4}}\right )}}{b x + a}\right ) - 3 i \, d \left (\frac {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}}\right )^{\frac {1}{4}} \log \left (-\frac {3 \, {\left ({\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} + {\left (-i \, b d^{2} x - i \, a d^{2}\right )} \left (\frac {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}}\right )^{\frac {1}{4}}\right )}}{b x + a}\right ) - 4 \, {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{4 \, d} \]
-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
\[ \int \frac {(a+b x)^{3/4}}{(c+d x)^{3/4}} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{4}}}{\left (c + d x\right )^{\frac {3}{4}}}\, dx \]
\[ \int \frac {(a+b x)^{3/4}}{(c+d x)^{3/4}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {3}{4}}}{{\left (d x + c\right )}^{\frac {3}{4}}} \,d x } \]
\[ \int \frac {(a+b x)^{3/4}}{(c+d x)^{3/4}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {3}{4}}}{{\left (d x + c\right )}^{\frac {3}{4}}} \,d x } \]
Timed out. \[ \int \frac {(a+b x)^{3/4}}{(c+d x)^{3/4}} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/4}}{{\left (c+d\,x\right )}^{3/4}} \,d x \]