Integrand size = 22, antiderivative size = 134 \[ \int \frac {1}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{a c x}+\frac {(3 b c+a d) \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{7/4} c^{5/4}}+\frac {(3 b c+a d) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{7/4} c^{5/4}} \]
-(b*x+a)^(1/4)*(d*x+c)^(3/4)/a/c/x+1/2*(a*d+3*b*c)*arctan(c^(1/4)*(b*x+a)^ (1/4)/a^(1/4)/(d*x+c)^(1/4))/a^(7/4)/c^(5/4)+1/2*(a*d+3*b*c)*arctanh(c^(1/ 4)*(b*x+a)^(1/4)/a^(1/4)/(d*x+c)^(1/4))/a^(7/4)/c^(5/4)
Time = 0.30 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\frac {-2 a^{3/4} \sqrt [4]{c} \sqrt [4]{a+b x} (c+d x)^{3/4}+(3 b c+a d) x \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )+(3 b c+a d) x \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{7/4} c^{5/4} x} \]
(-2*a^(3/4)*c^(1/4)*(a + b*x)^(1/4)*(c + d*x)^(3/4) + (3*b*c + a*d)*x*ArcT an[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))] + (3*b*c + a*d)*x* ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(2*a^(7/4)*c ^(5/4)*x)
Time = 0.23 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {107, 104, 756, 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.
| \(\displaystyle \int \frac {1}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx\) |
| \(\Big \downarrow \) 107 |
| \(\displaystyle -\frac {(a d+3 b c) \int \frac {1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}}dx}{4 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{a c x}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {(a d+3 b c) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}}{a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{a c x}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle -\frac {(a d+3 b c) \left (-\frac {\int \frac {1}{\sqrt {a}-\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c+d x}}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}}{2 \sqrt {a}}-\frac {\int \frac {1}{\sqrt {a}+\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c+d x}}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}}{2 \sqrt {a}}\right )}{a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{a c x}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {(a d+3 b c) \left (-\frac {\int \frac {1}{\sqrt {a}-\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c+d x}}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}}{2 \sqrt {a}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} \sqrt [4]{c}}\right )}{a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{a c x}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {(a d+3 b c) \left (-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} \sqrt [4]{c}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} \sqrt [4]{c}}\right )}{a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{a c x}\) |
-(((a + b*x)^(1/4)*(c + d*x)^(3/4))/(a*c*x)) - ((3*b*c + a*d)*(-1/2*ArcTan [(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))]/(a^(3/4)*c^(1/4)) - ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))]/(2*a^(3/4)*c^ (1/4))))/(a*c)
3.10.5.3.1 Defintions of rubi rules used
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
\[\int \frac {1}{x^{2} \left (b x +a \right )^{\frac {3}{4}} \left (d x +c \right )^{\frac {1}{4}}}d x\]
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 722, normalized size of antiderivative = 5.39 \[ \int \frac {1}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\frac {a c x \left (\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (3 \, b c + a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} + {\left (a^{2} c d x + a^{2} c^{2}\right )} \left (\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac {1}{4}}}{d x + c}\right ) - a c x \left (\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (3 \, b c + a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} - {\left (a^{2} c d x + a^{2} c^{2}\right )} \left (\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac {1}{4}}}{d x + c}\right ) - i \, a c x \left (\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (3 \, b c + a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} - {\left (i \, a^{2} c d x + i \, a^{2} c^{2}\right )} \left (\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac {1}{4}}}{d x + c}\right ) + i \, a c x \left (\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (3 \, b c + a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} - {\left (-i \, a^{2} c d x - i \, a^{2} c^{2}\right )} \left (\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac {1}{4}}}{d x + c}\right ) - 4 \, {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{4 \, a c x} \]
1/4*(a*c*x*((81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b* c*d^3 + a^4*d^4)/(a^7*c^5))^(1/4)*log(((3*b*c + a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (a^2*c*d*x + a^2*c^2)*((81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2 *b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*c^5))^(1/4))/(d*x + c)) - a* c*x*((81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*c^5))^(1/4)*log(((3*b*c + a*d)*(b*x + a)^(1/4)*(d*x + c)^(3 /4) - (a^2*c*d*x + a^2*c^2)*((81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^ 2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*c^5))^(1/4))/(d*x + c)) - I*a*c*x*( (81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4* d^4)/(a^7*c^5))^(1/4)*log(((3*b*c + a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (I*a^2*c*d*x + I*a^2*c^2)*((81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2 *d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*c^5))^(1/4))/(d*x + c)) + I*a*c*x*(( 81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d ^4)/(a^7*c^5))^(1/4)*log(((3*b*c + a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (-I*a^2*c*d*x - I*a^2*c^2)*((81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2 *d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*c^5))^(1/4))/(d*x + c)) - 4*(b*x + a )^(1/4)*(d*x + c)^(3/4))/(a*c*x)
\[ \int \frac {1}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int \frac {1}{x^{2} \left (a + b x\right )^{\frac {3}{4}} \sqrt [4]{c + d x}}\, dx \]
\[ \int \frac {1}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} x^{2}} \,d x } \]
\[ \int \frac {1}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} x^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int \frac {1}{x^2\,{\left (a+b\,x\right )}^{3/4}\,{\left (c+d\,x\right )}^{1/4}} \,d x \]