Integrand size = 22, antiderivative size = 288 \[ \int \frac {1}{x^4 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 a c x^3}+\frac {(11 b c+9 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{24 a^2 c^2 x^2}-\frac {\left (77 b^2 c^2+54 a b c d+45 a^2 d^2\right ) \sqrt [4]{a+b x} (c+d x)^{3/4}}{96 a^3 c^3 x}+\frac {\left (77 b^3 c^3+21 a b^2 c^2 d+15 a^2 b c d^2+15 a^3 d^3\right ) \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{64 a^{15/4} c^{13/4}}+\frac {\left (77 b^3 c^3+21 a b^2 c^2 d+15 a^2 b c d^2+15 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{64 a^{15/4} c^{13/4}} \]
-1/3*(b*x+a)^(1/4)*(d*x+c)^(3/4)/a/c/x^3+1/24*(9*a*d+11*b*c)*(b*x+a)^(1/4) *(d*x+c)^(3/4)/a^2/c^2/x^2-1/96*(45*a^2*d^2+54*a*b*c*d+77*b^2*c^2)*(b*x+a) ^(1/4)*(d*x+c)^(3/4)/a^3/c^3/x+1/64*(15*a^3*d^3+15*a^2*b*c*d^2+21*a*b^2*c^ 2*d+77*b^3*c^3)*arctan(c^(1/4)*(b*x+a)^(1/4)/a^(1/4)/(d*x+c)^(1/4))/a^(15/ 4)/c^(13/4)+1/64*(15*a^3*d^3+15*a^2*b*c*d^2+21*a*b^2*c^2*d+77*b^3*c^3)*arc tanh(c^(1/4)*(b*x+a)^(1/4)/a^(1/4)/(d*x+c)^(1/4))/a^(15/4)/c^(13/4)
Time = 0.84 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^4 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\frac {-\frac {2 a^{3/4} \sqrt [4]{c} \sqrt [4]{a+b x} (c+d x)^{3/4} \left (77 b^2 c^2 x^2+2 a b c x (-22 c+27 d x)+a^2 \left (32 c^2-36 c d x+45 d^2 x^2\right )\right )}{x^3}+3 \left (77 b^3 c^3+21 a b^2 c^2 d+15 a^2 b c d^2+15 a^3 d^3\right ) \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )+3 \left (77 b^3 c^3+21 a b^2 c^2 d+15 a^2 b c d^2+15 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{192 a^{15/4} c^{13/4}} \]
((-2*a^(3/4)*c^(1/4)*(a + b*x)^(1/4)*(c + d*x)^(3/4)*(77*b^2*c^2*x^2 + 2*a *b*c*x*(-22*c + 27*d*x) + a^2*(32*c^2 - 36*c*d*x + 45*d^2*x^2)))/x^3 + 3*( 77*b^3*c^3 + 21*a*b^2*c^2*d + 15*a^2*b*c*d^2 + 15*a^3*d^3)*ArcTan[(c^(1/4) *(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))] + 3*(77*b^3*c^3 + 21*a*b^2*c^ 2*d + 15*a^2*b*c*d^2 + 15*a^3*d^3)*ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1 /4)*(c + d*x)^(1/4))])/(192*a^(15/4)*c^(13/4))
Time = 0.37 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.95, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {114, 27, 168, 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.
| \(\displaystyle \int \frac {1}{x^4 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {\int \frac {11 b c+9 a d+8 b d x}{4 x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}}dx}{3 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 a c x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {11 b c+9 a d+8 b d x}{x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}}dx}{12 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 a c x^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {-\frac {\int \frac {77 b^2 c^2+54 a b d c+45 a^2 d^2+4 b d (11 b c+9 a d) x}{4 x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}}dx}{2 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (9 a d+11 b c)}{2 a c x^2}}{12 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 a c x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {77 b^2 c^2+54 a b d c+45 a^2 d^2+4 b d (11 b c+9 a d) x}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}}dx}{8 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (9 a d+11 b c)}{2 a c x^2}}{12 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 a c x^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {-\frac {-\frac {\int \frac {3 \left (77 b^3 c^3+21 a b^2 d c^2+15 a^2 b d^2 c+15 a^3 d^3\right )}{4 x (a+b x)^{3/4} \sqrt [4]{c+d x}}dx}{a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} \left (\frac {77 b^2 c}{a}+\frac {45 a d^2}{c}+54 b d\right )}{x}}{8 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (9 a d+11 b c)}{2 a c x^2}}{12 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 a c x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {-\frac {3 \left (15 a^3 d^3+15 a^2 b c d^2+21 a b^2 c^2 d+77 b^3 c^3\right ) \int \frac {1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}}dx}{4 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} \left (\frac {77 b^2 c}{a}+\frac {45 a d^2}{c}+54 b d\right )}{x}}{8 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (9 a d+11 b c)}{2 a c x^2}}{12 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 a c x^3}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {-\frac {-\frac {3 \left (15 a^3 d^3+15 a^2 b c d^2+21 a b^2 c^2 d+77 b^3 c^3\right ) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}}{a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} \left (\frac {77 b^2 c}{a}+\frac {45 a d^2}{c}+54 b d\right )}{x}}{8 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (9 a d+11 b c)}{2 a c x^2}}{12 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 a c x^3}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle -\frac {-\frac {-\frac {3 \left (15 a^3 d^3+15 a^2 b c d^2+21 a b^2 c^2 d+77 b^3 c^3\right ) \left (-\frac {\int \frac {1}{\sqrt {a}-\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c+d x}}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}}{2 \sqrt {a}}-\frac {\int \frac {1}{\sqrt {a}+\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c+d x}}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}}{2 \sqrt {a}}\right )}{a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} \left (\frac {77 b^2 c}{a}+\frac {45 a d^2}{c}+54 b d\right )}{x}}{8 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (9 a d+11 b c)}{2 a c x^2}}{12 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 a c x^3}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {-\frac {-\frac {3 \left (15 a^3 d^3+15 a^2 b c d^2+21 a b^2 c^2 d+77 b^3 c^3\right ) \left (-\frac {\int \frac {1}{\sqrt {a}-\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c+d x}}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}}{2 \sqrt {a}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} \sqrt [4]{c}}\right )}{a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} \left (\frac {77 b^2 c}{a}+\frac {45 a d^2}{c}+54 b d\right )}{x}}{8 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (9 a d+11 b c)}{2 a c x^2}}{12 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 a c x^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {-\frac {-\frac {3 \left (15 a^3 d^3+15 a^2 b c d^2+21 a b^2 c^2 d+77 b^3 c^3\right ) \left (-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} \sqrt [4]{c}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} \sqrt [4]{c}}\right )}{a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} \left (\frac {77 b^2 c}{a}+\frac {45 a d^2}{c}+54 b d\right )}{x}}{8 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (9 a d+11 b c)}{2 a c x^2}}{12 a c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 a c x^3}\) |
-1/3*((a + b*x)^(1/4)*(c + d*x)^(3/4))/(a*c*x^3) - (-1/2*((11*b*c + 9*a*d) *(a + b*x)^(1/4)*(c + d*x)^(3/4))/(a*c*x^2) - (-((((77*b^2*c)/a + 54*b*d + (45*a*d^2)/c)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/x) - (3*(77*b^3*c^3 + 21*a *b^2*c^2*d + 15*a^2*b*c*d^2 + 15*a^3*d^3)*(-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))/(12*a*c)
3.10.7.3.1 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]
\[\int \frac {1}{x^{4} \left (b x +a \right )^{\frac {3}{4}} \left (d x +c \right )^{\frac {1}{4}}}d x\]
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 1829, normalized size of antiderivative = 6.35 \[ \int \frac {1}{x^4 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\text {Too large to display} \]
1/384*(3*a^3*c^3*x^3*((35153041*b^12*c^12 + 38348772*a*b^11*c^11*d + 43080 114*a^2*b^10*c^10*d^2 + 52655988*a^3*b^9*c^9*d^3 + 36722511*a^4*b^8*c^8*d^ 4 + 27042120*a^5*b^7*c^7*d^5 + 18926460*a^6*b^6*c^6*d^6 + 8958600*a^7*b^5* c^5*d^7 + 4614975*a^8*b^4*c^4*d^8 + 2092500*a^9*b^3*c^3*d^9 + 587250*a^10* b^2*c^2*d^10 + 202500*a^11*b*c*d^11 + 50625*a^12*d^12)/(a^15*c^13))^(1/4)* log(((77*b^3*c^3 + 21*a*b^2*c^2*d + 15*a^2*b*c*d^2 + 15*a^3*d^3)*(b*x + a) ^(1/4)*(d*x + c)^(3/4) + (a^4*c^3*d*x + a^4*c^4)*((35153041*b^12*c^12 + 38 348772*a*b^11*c^11*d + 43080114*a^2*b^10*c^10*d^2 + 52655988*a^3*b^9*c^9*d ^3 + 36722511*a^4*b^8*c^8*d^4 + 27042120*a^5*b^7*c^7*d^5 + 18926460*a^6*b^ 6*c^6*d^6 + 8958600*a^7*b^5*c^5*d^7 + 4614975*a^8*b^4*c^4*d^8 + 2092500*a^ 9*b^3*c^3*d^9 + 587250*a^10*b^2*c^2*d^10 + 202500*a^11*b*c*d^11 + 50625*a^ 12*d^12)/(a^15*c^13))^(1/4))/(d*x + c)) - 3*a^3*c^3*x^3*((35153041*b^12*c^ 12 + 38348772*a*b^11*c^11*d + 43080114*a^2*b^10*c^10*d^2 + 52655988*a^3*b^ 9*c^9*d^3 + 36722511*a^4*b^8*c^8*d^4 + 27042120*a^5*b^7*c^7*d^5 + 18926460 *a^6*b^6*c^6*d^6 + 8958600*a^7*b^5*c^5*d^7 + 4614975*a^8*b^4*c^4*d^8 + 209 2500*a^9*b^3*c^3*d^9 + 587250*a^10*b^2*c^2*d^10 + 202500*a^11*b*c*d^11 + 5 0625*a^12*d^12)/(a^15*c^13))^(1/4)*log(((77*b^3*c^3 + 21*a*b^2*c^2*d + 15* a^2*b*c*d^2 + 15*a^3*d^3)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (a^4*c^3*d*x + a^4*c^4)*((35153041*b^12*c^12 + 38348772*a*b^11*c^11*d + 43080114*a^2*b^1 0*c^10*d^2 + 52655988*a^3*b^9*c^9*d^3 + 36722511*a^4*b^8*c^8*d^4 + 2704...
\[ \int \frac {1}{x^4 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int \frac {1}{x^{4} \left (a + b x\right )^{\frac {3}{4}} \sqrt [4]{c + d x}}\, dx \]
\[ \int \frac {1}{x^4 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} x^{4}} \,d x } \]
\[ \int \frac {1}{x^4 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} x^{4}} \,d x } \]
Timed out. \[ \int \frac {1}{x^4 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int \frac {1}{x^4\,{\left (a+b\,x\right )}^{3/4}\,{\left (c+d\,x\right )}^{1/4}} \,d x \]