Integrand size = 22, antiderivative size = 340 \[ \int \frac {x^3 \sqrt [4]{a+b x}}{\sqrt [4]{c+d x}} \, dx=-\frac {\left (195 b^3 c^3+135 a b^2 c^2 d+105 a^2 b c d^2+77 a^3 d^3\right ) \sqrt [4]{a+b x} (c+d x)^{3/4}}{512 b^3 d^4}+\frac {x^2 (a+b x)^{5/4} (c+d x)^{3/4}}{4 b d}+\frac {(a+b x)^{5/4} (c+d x)^{3/4} \left (117 b^2 c^2+94 a b c d+77 a^2 d^2-8 b d (13 b c+11 a d) x\right )}{384 b^3 d^3}+\frac {(b c-a d) \left (195 b^3 c^3+135 a b^2 c^2 d+105 a^2 b c d^2+77 a^3 d^3\right ) \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{1024 b^{15/4} d^{17/4}}+\frac {(b c-a d) \left (195 b^3 c^3+135 a b^2 c^2 d+105 a^2 b c d^2+77 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{1024 b^{15/4} d^{17/4}} \]
-1/512*(77*a^3*d^3+105*a^2*b*c*d^2+135*a*b^2*c^2*d+195*b^3*c^3)*(b*x+a)^(1 /4)*(d*x+c)^(3/4)/b^3/d^4+1/4*x^2*(b*x+a)^(5/4)*(d*x+c)^(3/4)/b/d+1/384*(b *x+a)^(5/4)*(d*x+c)^(3/4)*(117*b^2*c^2+94*a*b*c*d+77*a^2*d^2-8*b*d*(11*a*d +13*b*c)*x)/b^3/d^3+1/1024*(-a*d+b*c)*(77*a^3*d^3+105*a^2*b*c*d^2+135*a*b^ 2*c^2*d+195*b^3*c^3)*arctan(d^(1/4)*(b*x+a)^(1/4)/b^(1/4)/(d*x+c)^(1/4))/b ^(15/4)/d^(17/4)+1/1024*(-a*d+b*c)*(77*a^3*d^3+105*a^2*b*c*d^2+135*a*b^2*c ^2*d+195*b^3*c^3)*arctanh(d^(1/4)*(b*x+a)^(1/4)/b^(1/4)/(d*x+c)^(1/4))/b^( 15/4)/d^(17/4)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.46 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.69 \[ \int \frac {x^3 \sqrt [4]{a+b x}}{\sqrt [4]{c+d x}} \, dx=\frac {(a+b x)^{5/4} \sqrt [4]{\frac {b (c+d x)}{b c-a d}} \left (-(b c-a d)^2 (13 b c+11 a d) \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},\frac {5}{4},\frac {9}{4},\frac {d (a+b x)}{-b c+a d}\right )+b \left (2 c \left (13 b^2 c^2-6 a b c d-7 a^2 d^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {5}{4},\frac {9}{4},\frac {d (a+b x)}{-b c+a d}\right )+b \left (\frac {5 b d^2 x^2 (c+d x)}{\sqrt [4]{\frac {b (c+d x)}{b c-a d}}}-c^2 (13 b c+3 a d) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {5}{4},\frac {9}{4},\frac {d (a+b x)}{-b c+a d}\right )\right )\right )\right )}{20 b^4 d^3 \sqrt [4]{c+d x}} \]
((a + b*x)^(5/4)*((b*(c + d*x))/(b*c - a*d))^(1/4)*(-((b*c - a*d)^2*(13*b* c + 11*a*d)*Hypergeometric2F1[-7/4, 5/4, 9/4, (d*(a + b*x))/(-(b*c) + a*d) ]) + b*(2*c*(13*b^2*c^2 - 6*a*b*c*d - 7*a^2*d^2)*Hypergeometric2F1[-3/4, 5 /4, 9/4, (d*(a + b*x))/(-(b*c) + a*d)] + b*((5*b*d^2*x^2*(c + d*x))/((b*(c + d*x))/(b*c - a*d))^(1/4) - c^2*(13*b*c + 3*a*d)*Hypergeometric2F1[1/4, 5/4, 9/4, (d*(a + b*x))/(-(b*c) + a*d)]))))/(20*b^4*d^3*(c + d*x)^(1/4))
Time = 0.39 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.91, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {111, 27, 164, 60, 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.
| \(\displaystyle \int \frac {x^3 \sqrt [4]{a+b x}}{\sqrt [4]{c+d x}} \, dx\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {\int -\frac {x \sqrt [4]{a+b x} (8 a c+(13 b c+11 a d) x)}{4 \sqrt [4]{c+d x}}dx}{4 b d}+\frac {x^2 (a+b x)^{5/4} (c+d x)^{3/4}}{4 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^2 (a+b x)^{5/4} (c+d x)^{3/4}}{4 b d}-\frac {\int \frac {x \sqrt [4]{a+b x} (8 a c+(13 b c+11 a d) x)}{\sqrt [4]{c+d x}}dx}{16 b d}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {x^2 (a+b x)^{5/4} (c+d x)^{3/4}}{4 b d}-\frac {\frac {\left (77 a^3 d^3+105 a^2 b c d^2+135 a b^2 c^2 d+195 b^3 c^3\right ) \int \frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}dx}{32 b^2 d^2}-\frac {(a+b x)^{5/4} (c+d x)^{3/4} \left (77 a^2 d^2-8 b d x (11 a d+13 b c)+94 a b c d+117 b^2 c^2\right )}{24 b^2 d^2}}{16 b d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {x^2 (a+b x)^{5/4} (c+d x)^{3/4}}{4 b d}-\frac {\frac {\left (77 a^3 d^3+105 a^2 b c d^2+135 a b^2 c^2 d+195 b^3 c^3\right ) \left (\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{d}-\frac {(b c-a d) \int \frac {1}{(a+b x)^{3/4} \sqrt [4]{c+d x}}dx}{4 d}\right )}{32 b^2 d^2}-\frac {(a+b x)^{5/4} (c+d x)^{3/4} \left (77 a^2 d^2-8 b d x (11 a d+13 b c)+94 a b c d+117 b^2 c^2\right )}{24 b^2 d^2}}{16 b d}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {x^2 (a+b x)^{5/4} (c+d x)^{3/4}}{4 b d}-\frac {\frac {\left (77 a^3 d^3+105 a^2 b c d^2+135 a b^2 c^2 d+195 b^3 c^3\right ) \left (\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}d\sqrt [4]{a+b x}}{b d}\right )}{32 b^2 d^2}-\frac {(a+b x)^{5/4} (c+d x)^{3/4} \left (77 a^2 d^2-8 b d x (11 a d+13 b c)+94 a b c d+117 b^2 c^2\right )}{24 b^2 d^2}}{16 b d}\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle \frac {x^2 (a+b x)^{5/4} (c+d x)^{3/4}}{4 b d}-\frac {\frac {\left (77 a^3 d^3+105 a^2 b c d^2+135 a b^2 c^2 d+195 b^3 c^3\right ) \left (\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{d}-\frac {(b c-a d) \int \frac {1}{1-\frac {d (a+b x)}{b}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{b d}\right )}{32 b^2 d^2}-\frac {(a+b x)^{5/4} (c+d x)^{3/4} \left (77 a^2 d^2-8 b d x (11 a d+13 b c)+94 a b c d+117 b^2 c^2\right )}{24 b^2 d^2}}{16 b d}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {x^2 (a+b x)^{5/4} (c+d x)^{3/4}}{4 b d}-\frac {\frac {\left (77 a^3 d^3+105 a^2 b c d^2+135 a b^2 c^2 d+195 b^3 c^3\right ) \left (\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{d}-\frac {(b c-a d) \left (\frac {1}{2} \sqrt {b} \int \frac {1}{\sqrt {b}-\sqrt {d} \sqrt {a+b x}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {1}{2} \sqrt {b} \int \frac {1}{\sqrt {b}+\sqrt {d} \sqrt {a+b x}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}\right )}{b d}\right )}{32 b^2 d^2}-\frac {(a+b x)^{5/4} (c+d x)^{3/4} \left (77 a^2 d^2-8 b d x (11 a d+13 b c)+94 a b c d+117 b^2 c^2\right )}{24 b^2 d^2}}{16 b d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {x^2 (a+b x)^{5/4} (c+d x)^{3/4}}{4 b d}-\frac {\frac {\left (77 a^3 d^3+105 a^2 b c d^2+135 a b^2 c^2 d+195 b^3 c^3\right ) \left (\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{d}-\frac {(b c-a d) \left (\frac {1}{2} \sqrt {b} \int \frac {1}{\sqrt {b}-\sqrt {d} \sqrt {a+b x}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}+\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{2 \sqrt [4]{d}}\right )}{b d}\right )}{32 b^2 d^2}-\frac {(a+b x)^{5/4} (c+d x)^{3/4} \left (77 a^2 d^2-8 b d x (11 a d+13 b c)+94 a b c d+117 b^2 c^2\right )}{24 b^2 d^2}}{16 b d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {x^2 (a+b x)^{5/4} (c+d x)^{3/4}}{4 b d}-\frac {\frac {\left (77 a^3 d^3+105 a^2 b c d^2+135 a b^2 c^2 d+195 b^3 c^3\right ) \left (\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{d}-\frac {(b c-a d) \left (\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{2 \sqrt [4]{d}}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{2 \sqrt [4]{d}}\right )}{b d}\right )}{32 b^2 d^2}-\frac {(a+b x)^{5/4} (c+d x)^{3/4} \left (77 a^2 d^2-8 b d x (11 a d+13 b c)+94 a b c d+117 b^2 c^2\right )}{24 b^2 d^2}}{16 b d}\) |
(x^2*(a + b*x)^(5/4)*(c + d*x)^(3/4))/(4*b*d) - (-1/24*((a + b*x)^(5/4)*(c + d*x)^(3/4)*(117*b^2*c^2 + 94*a*b*c*d + 77*a^2*d^2 - 8*b*d*(13*b*c + 11* a*d)*x))/(b^2*d^2) + ((195*b^3*c^3 + 135*a*b^2*c^2*d + 105*a^2*b*c*d^2 + 7 7*a^3*d^3)*(((a + b*x)^(1/4)*(c + d*x)^(3/4))/d - ((b*c - a*d)*((b^(1/4)*A rcTan[(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*d)))/(32*b^2*d^2 ))/(16*b*d)
3.9.83.3.1 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_))^(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 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ ))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h *(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)) Int[( a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 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]
\[\int \frac {x^{3} \left (b x +a \right )^{\frac {1}{4}}}{\left (d x +c \right )^{\frac {1}{4}}}d x\]
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 2370, normalized size of antiderivative = 6.97 \[ \int \frac {x^3 \sqrt [4]{a+b x}}{\sqrt [4]{c+d x}} \, dx=\text {Too large to display} \]
1/6144*(3*b^3*d^4*((1445900625*b^16*c^16 - 1779570000*a*b^15*c^15*d - 6844 5000*a^2*b^14*c^14*d^2 - 177606000*a^3*b^13*c^13*d^3 - 1551622500*a^4*b^12 *c^12*d^4 + 2155086000*a^5*b^11*c^11*d^5 + 370974600*a^6*b^10*c^10*d^6 + 2 75389200*a^7*b^9*c^9*d^7 + 665778150*a^8*b^8*c^8*d^8 - 989262960*a^9*b^7*c ^7*d^9 - 318453240*a^10*b^6*c^6*d^10 - 191017680*a^11*b^5*c^5*d^11 - 18220 3364*a^12*b^4*c^4*d^12 + 176093456*a^13*b^3*c^3*d^13 + 82673976*a^14*b^2*c ^2*d^14 + 51131696*a^15*b*c*d^15 + 35153041*a^16*d^16)/(b^15*d^17))^(1/4)* log(-((195*b^4*c^4 - 60*a*b^3*c^3*d - 30*a^2*b^2*c^2*d^2 - 28*a^3*b*c*d^3 - 77*a^4*d^4)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (b^4*d^5*x + b^4*c*d^4)*(( 1445900625*b^16*c^16 - 1779570000*a*b^15*c^15*d - 68445000*a^2*b^14*c^14*d ^2 - 177606000*a^3*b^13*c^13*d^3 - 1551622500*a^4*b^12*c^12*d^4 + 21550860 00*a^5*b^11*c^11*d^5 + 370974600*a^6*b^10*c^10*d^6 + 275389200*a^7*b^9*c^9 *d^7 + 665778150*a^8*b^8*c^8*d^8 - 989262960*a^9*b^7*c^7*d^9 - 318453240*a ^10*b^6*c^6*d^10 - 191017680*a^11*b^5*c^5*d^11 - 182203364*a^12*b^4*c^4*d^ 12 + 176093456*a^13*b^3*c^3*d^13 + 82673976*a^14*b^2*c^2*d^14 + 51131696*a ^15*b*c*d^15 + 35153041*a^16*d^16)/(b^15*d^17))^(1/4))/(d*x + c)) - 3*b^3* d^4*((1445900625*b^16*c^16 - 1779570000*a*b^15*c^15*d - 68445000*a^2*b^14* c^14*d^2 - 177606000*a^3*b^13*c^13*d^3 - 1551622500*a^4*b^12*c^12*d^4 + 21 55086000*a^5*b^11*c^11*d^5 + 370974600*a^6*b^10*c^10*d^6 + 275389200*a^7*b ^9*c^9*d^7 + 665778150*a^8*b^8*c^8*d^8 - 989262960*a^9*b^7*c^7*d^9 - 31...
\[ \int \frac {x^3 \sqrt [4]{a+b x}}{\sqrt [4]{c+d x}} \, dx=\int \frac {x^{3} \sqrt [4]{a + b x}}{\sqrt [4]{c + d x}}\, dx \]
\[ \int \frac {x^3 \sqrt [4]{a+b x}}{\sqrt [4]{c+d x}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{4}} x^{3}}{{\left (d x + c\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {x^3 \sqrt [4]{a+b x}}{\sqrt [4]{c+d x}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{4}} x^{3}}{{\left (d x + c\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {x^3 \sqrt [4]{a+b x}}{\sqrt [4]{c+d x}} \, dx=\int \frac {x^3\,{\left (a+b\,x\right )}^{1/4}}{{\left (c+d\,x\right )}^{1/4}} \,d x \]