Integrand size = 22, antiderivative size = 201 \[ \int \frac {x^2}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=-\frac {(5 b c+7 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 b^2 d^2}+\frac {x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{11/4} d^{9/4}}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{11/4} d^{9/4}} \]
-1/8*(7*a*d+5*b*c)*(b*x+a)^(1/4)*(d*x+c)^(3/4)/b^2/d^2+1/2*x*(b*x+a)^(1/4) *(d*x+c)^(3/4)/b/d+1/16*(21*a^2*d^2+6*a*b*c*d+5*b^2*c^2)*arctan(d^(1/4)*(b *x+a)^(1/4)/b^(1/4)/(d*x+c)^(1/4))/b^(11/4)/d^(9/4)+1/16*(21*a^2*d^2+6*a*b *c*d+5*b^2*c^2)*arctanh(d^(1/4)*(b*x+a)^(1/4)/b^(1/4)/(d*x+c)^(1/4))/b^(11 /4)/d^(9/4)
Time = 0.54 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.83 \[ \int \frac {x^2}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\frac {2 b^{3/4} \sqrt [4]{d} \sqrt [4]{a+b x} (c+d x)^{3/4} (-5 b c-7 a d+4 b d x)+\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )+\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{11/4} d^{9/4}} \]
(2*b^(3/4)*d^(1/4)*(a + b*x)^(1/4)*(c + d*x)^(3/4)*(-5*b*c - 7*a*d + 4*b*d *x) + (5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*ArcTan[(d^(1/4)*(a + b*x)^(1/4) )/(b^(1/4)*(c + d*x)^(1/4))] + (5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*ArcTan h[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(16*b^(11/4)*d^(9/ 4))
Time = 0.32 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {101, 27, 90, 73, 770, 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.
| \(\displaystyle \int \frac {x^2}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle \frac {\int -\frac {4 a c+(5 b c+7 a d) x}{4 (a+b x)^{3/4} \sqrt [4]{c+d x}}dx}{2 b d}+\frac {x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d}-\frac {\int \frac {4 a c+(5 b c+7 a d) x}{(a+b x)^{3/4} \sqrt [4]{c+d x}}dx}{8 b d}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d}-\frac {\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (7 a d+5 b c)}{b d}-\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \int \frac {1}{(a+b x)^{3/4} \sqrt [4]{c+d x}}dx}{4 b d}}{8 b d}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d}-\frac {\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (7 a d+5 b c)}{b d}-\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \int \frac {1}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}d\sqrt [4]{a+b x}}{b^2 d}}{8 b d}\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle \frac {x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d}-\frac {\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (7 a d+5 b c)}{b d}-\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \int \frac {1}{1-\frac {d (a+b x)}{b}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{b^2 d}}{8 b d}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d}-\frac {\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (7 a d+5 b c)}{b d}-\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \left (\frac {1}{2} \sqrt {b} \int \frac {1}{\sqrt {b}-\sqrt {d} \sqrt {a+b x}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {1}{2} \sqrt {b} \int \frac {1}{\sqrt {b}+\sqrt {d} \sqrt {a+b x}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{b^2 d}}{8 b d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d}-\frac {\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (7 a d+5 b c)}{b d}-\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \left (\frac {1}{2} \sqrt {b} \int \frac {1}{\sqrt {b}-\sqrt {d} \sqrt {a+b x}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{2 \sqrt [4]{d}}\right )}{b^2 d}}{8 b d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d}-\frac {\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (7 a d+5 b c)}{b d}-\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \left (\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{2 \sqrt [4]{d}}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{2 \sqrt [4]{d}}\right )}{b^2 d}}{8 b d}\) |
(x*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(2*b*d) - (((5*b*c + 7*a*d)*(a + b*x)^ (1/4)*(c + d*x)^(3/4))/(b*d) - ((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*((b^( 1/4)*ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c - (a*d)/b + (d*(a + b*x) )/b)^(1/4))])/(2*d^(1/4)) + (b^(1/4)*ArcTanh[(d^(1/4)*(a + b*x)^(1/4))/(b^ (1/4)*(c - (a*d)/b + (d*(a + b*x))/b)^(1/4))])/(2*d^(1/4))))/(b^2*d))/(8*b *d)
3.10.1.3.1 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] :> 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
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[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n) Subst[In t[1/(1 - b*x^n)^(p + 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)] && IntegerQ[p + 1 /n]
\[\int \frac {x^{2}}{\left (b x +a \right )^{\frac {3}{4}} \left (d x +c \right )^{\frac {1}{4}}}d x\]
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 1269, normalized size of antiderivative = 6.31 \[ \int \frac {x^2}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\text {Too large to display} \]
1/32*(b^2*d^2*((625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 4 2120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 176904*a^5*b^3*c^3*d^5 + 2 80476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^8)/(b^11*d^9))^( 1/4)*log(((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*(b*x + a)^(1/4)*(d*x + c)^( 3/4) + (b^3*d^3*x + b^3*c*d^2)*((625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^ 2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 176904*a^ 5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d ^8)/(b^11*d^9))^(1/4))/(d*x + c)) - b^2*d^2*((625*b^8*c^8 + 3000*a*b^7*c^7 *d + 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^ 4 + 176904*a^5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^8)/(b^11*d^9))^(1/4)*log(((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^ 2)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (b^3*d^3*x + b^3*c*d^2)*((625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 1128 06*a^4*b^4*c^4*d^4 + 176904*a^5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222 264*a^7*b*c*d^7 + 194481*a^8*d^8)/(b^11*d^9))^(1/4))/(d*x + c)) - I*b^2*d^ 2*((625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5 *c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 176904*a^5*b^3*c^3*d^5 + 280476*a^6*b^ 2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^8)/(b^11*d^9))^(1/4)*log(((5 *b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (I*b^ 3*d^3*x + I*b^3*c*d^2)*((625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6...
\[ \int \frac {x^2}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int \frac {x^{2}}{\left (a + b x\right )^{\frac {3}{4}} \sqrt [4]{c + d x}}\, dx \]
\[ \int \frac {x^2}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int { \frac {x^{2}}{{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {x^2}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int { \frac {x^{2}}{{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {x^2}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int \frac {x^2}{{\left (a+b\,x\right )}^{3/4}\,{\left (c+d\,x\right )}^{1/4}} \,d x \]