Integrand size = 22, antiderivative size = 206 \[ \int \frac {1}{x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2}+\frac {(7 b c+5 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 a^2 c^2 x}-\frac {\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{11/4} c^{9/4}}-\frac {\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{11/4} c^{9/4}} \]
-1/2*(b*x+a)^(1/4)*(d*x+c)^(3/4)/a/c/x^2+1/8*(5*a*d+7*b*c)*(b*x+a)^(1/4)*( d*x+c)^(3/4)/a^2/c^2/x-1/16*(5*a^2*d^2+6*a*b*c*d+21*b^2*c^2)*arctan(c^(1/4 )*(b*x+a)^(1/4)/a^(1/4)/(d*x+c)^(1/4))/a^(11/4)/c^(9/4)-1/16*(5*a^2*d^2+6* a*b*c*d+21*b^2*c^2)*arctanh(c^(1/4)*(b*x+a)^(1/4)/a^(1/4)/(d*x+c)^(1/4))/a ^(11/4)/c^(9/4)
Time = 0.59 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\frac {2 a^{3/4} \sqrt [4]{c} \sqrt [4]{a+b x} (c+d x)^{3/4} (-4 a c+7 b c x+5 a d x)-\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) x^2 \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )-\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) x^2 \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{11/4} c^{9/4} x^2} \]
(2*a^(3/4)*c^(1/4)*(a + b*x)^(1/4)*(c + d*x)^(3/4)*(-4*a*c + 7*b*c*x + 5*a *d*x) - (21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*x^2*ArcTan[(c^(1/4)*(a + b*x) ^(1/4))/(a^(1/4)*(c + d*x)^(1/4))] - (21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)* x^2*ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(16*a^(1 1/4)*c^(9/4)*x^2)
Time = 0.28 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.98, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {114, 27, 168, 27, 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.
| \(\displaystyle \int \frac {1}{x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {\int \frac {7 b c+5 a d+4 b d x}{4 x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}}dx}{2 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {7 b c+5 a d+4 b d x}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}}dx}{8 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {-\frac {\int \frac {21 b^2 c^2+6 a b d c+5 a^2 d^2}{4 x (a+b x)^{3/4} \sqrt [4]{c+d x}}dx}{a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (5 a d+7 b c)}{a c x}}{8 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \int \frac {1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}}dx}{4 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (5 a d+7 b c)}{a c x}}{8 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {-\frac {\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}}{a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (5 a d+7 b c)}{a c x}}{8 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle -\frac {-\frac {\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \left (-\frac {\int \frac {1}{\sqrt {a}-\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c+d x}}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}}{2 \sqrt {a}}-\frac {\int \frac {1}{\sqrt {a}+\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c+d x}}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}}{2 \sqrt {a}}\right )}{a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (5 a d+7 b c)}{a c x}}{8 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {-\frac {\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \left (-\frac {\int \frac {1}{\sqrt {a}-\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c+d x}}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}}{2 \sqrt {a}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} \sqrt [4]{c}}\right )}{a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (5 a d+7 b c)}{a c x}}{8 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {-\frac {\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \left (-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} \sqrt [4]{c}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} \sqrt [4]{c}}\right )}{a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (5 a d+7 b c)}{a c x}}{8 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2}\) |
-1/2*((a + b*x)^(1/4)*(c + d*x)^(3/4))/(a*c*x^2) - (-(((7*b*c + 5*a*d)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(a*c*x)) - ((21*b^2*c^2 + 6*a*b*c*d + 5*a^2* d^2)*(-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))/(8*a*c)
3.10.6.3.1 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_))/((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[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*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(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] + S imp[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*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[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]
\[\int \frac {1}{x^{3} \left (b x +a \right )^{\frac {3}{4}} \left (d x +c \right )^{\frac {1}{4}}}d x\]
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 1287, normalized size of antiderivative = 6.25 \[ \int \frac {1}{x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\text {Too large to display} \]
-1/32*(a^2*c^2*x^2*((194481*b^8*c^8 + 222264*a*b^7*c^7*d + 280476*a^2*b^6* c^6*d^2 + 176904*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 42120*a^5*b^3* c^3*d^5 + 15900*a^6*b^2*c^2*d^6 + 3000*a^7*b*c*d^7 + 625*a^8*d^8)/(a^11*c^ 9))^(1/4)*log(((21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (a^3*c^2*d*x + a^3*c^3)*((194481*b^8*c^8 + 222264*a*b^7*c^7*d + 280476*a^2*b^6*c^6*d^2 + 176904*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 42120*a^5*b^3*c^3*d^5 + 15900*a^6*b^2*c^2*d^6 + 3000*a^7*b*c*d^7 + 625* a^8*d^8)/(a^11*c^9))^(1/4))/(d*x + c)) - a^2*c^2*x^2*((194481*b^8*c^8 + 22 2264*a*b^7*c^7*d + 280476*a^2*b^6*c^6*d^2 + 176904*a^3*b^5*c^5*d^3 + 11280 6*a^4*b^4*c^4*d^4 + 42120*a^5*b^3*c^3*d^5 + 15900*a^6*b^2*c^2*d^6 + 3000*a ^7*b*c*d^7 + 625*a^8*d^8)/(a^11*c^9))^(1/4)*log(((21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (a^3*c^2*d*x + a^3*c^3)*((19 4481*b^8*c^8 + 222264*a*b^7*c^7*d + 280476*a^2*b^6*c^6*d^2 + 176904*a^3*b^ 5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 42120*a^5*b^3*c^3*d^5 + 15900*a^6*b^2 *c^2*d^6 + 3000*a^7*b*c*d^7 + 625*a^8*d^8)/(a^11*c^9))^(1/4))/(d*x + c)) - I*a^2*c^2*x^2*((194481*b^8*c^8 + 222264*a*b^7*c^7*d + 280476*a^2*b^6*c^6* d^2 + 176904*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 42120*a^5*b^3*c^3* d^5 + 15900*a^6*b^2*c^2*d^6 + 3000*a^7*b*c*d^7 + 625*a^8*d^8)/(a^11*c^9))^ (1/4)*log(((21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*(b*x + a)^(1/4)*(d*x + c)^ (3/4) - (I*a^3*c^2*d*x + I*a^3*c^3)*((194481*b^8*c^8 + 222264*a*b^7*c^7...
\[ \int \frac {1}{x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int \frac {1}{x^{3} \left (a + b x\right )^{\frac {3}{4}} \sqrt [4]{c + d x}}\, dx \]
\[ \int \frac {1}{x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} x^{3}} \,d x } \]
\[ \int \frac {1}{x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} x^{3}} \,d x } \]
Timed out. \[ \int \frac {1}{x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int \frac {1}{x^3\,{\left (a+b\,x\right )}^{3/4}\,{\left (c+d\,x\right )}^{1/4}} \,d x \]