Integrand size = 19, antiderivative size = 167 \[ \int \frac {(a+b x)^{7/4}}{(c+d x)^{3/4}} \, dx=-\frac {7 (b c-a d) (a+b x)^{3/4} \sqrt [4]{c+d x}}{8 d^2}+\frac {(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 d}-\frac {21 (b c-a d)^2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 \sqrt [4]{b} d^{11/4}}+\frac {21 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 \sqrt [4]{b} d^{11/4}} \]
-7/8*(-a*d+b*c)*(b*x+a)^(3/4)*(d*x+c)^(1/4)/d^2+1/2*(b*x+a)^(7/4)*(d*x+c)^ (1/4)/d-21/16*(-a*d+b*c)^2*arctan(d^(1/4)*(b*x+a)^(1/4)/b^(1/4)/(d*x+c)^(1 /4))/b^(1/4)/d^(11/4)+21/16*(-a*d+b*c)^2*arctanh(d^(1/4)*(b*x+a)^(1/4)/b^( 1/4)/(d*x+c)^(1/4))/b^(1/4)/d^(11/4)
Time = 0.32 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x)^{7/4}}{(c+d x)^{3/4}} \, dx=\frac {2 d^{3/4} (a+b x)^{3/4} \sqrt [4]{c+d x} (-7 b c+11 a d+4 b d x)+\frac {21 (b c-a d)^2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )}{\sqrt [4]{b}}+\frac {21 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d} \sqrt [4]{a+b x}}\right )}{\sqrt [4]{b}}}{16 d^{11/4}} \]
(2*d^(3/4)*(a + b*x)^(3/4)*(c + d*x)^(1/4)*(-7*b*c + 11*a*d + 4*b*d*x) + ( 21*(b*c - a*d)^2*ArcTan[(b^(1/4)*(c + d*x)^(1/4))/(d^(1/4)*(a + b*x)^(1/4) )])/b^(1/4) + (21*(b*c - a*d)^2*ArcTanh[(b^(1/4)*(c + d*x)^(1/4))/(d^(1/4) *(a + b*x)^(1/4))])/b^(1/4))/(16*d^(11/4))
Time = 0.28 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {60, 60, 73, 854, 27, 827, 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.
| \(\displaystyle \int \frac {(a+b x)^{7/4}}{(c+d x)^{3/4}} \, dx\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 d}-\frac {7 (b c-a d) \int \frac {(a+b x)^{3/4}}{(c+d x)^{3/4}}dx}{8 d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 d}-\frac {7 (b c-a d) \left (\frac {(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}-\frac {3 (b c-a d) \int \frac {1}{\sqrt [4]{a+b x} (c+d x)^{3/4}}dx}{4 d}\right )}{8 d}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 d}-\frac {7 (b c-a d) \left (\frac {(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}-\frac {3 (b c-a d) \int \frac {\sqrt {a+b x}}{\left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right )^{3/4}}d\sqrt [4]{a+b x}}{b d}\right )}{8 d}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle \frac {(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 d}-\frac {7 (b c-a d) \left (\frac {(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}-\frac {3 (b c-a d) \int \frac {b \sqrt {a+b x}}{b-d (a+b x)}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{b d}\right )}{8 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 d}-\frac {7 (b c-a d) \left (\frac {(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}-\frac {3 (b c-a d) \int \frac {\sqrt {a+b x}}{b-d (a+b x)}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{d}\right )}{8 d}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 d}-\frac {7 (b c-a d) \left (\frac {(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}-\frac {3 (b c-a d) \left (\frac {\int \frac {1}{\sqrt {b}-\sqrt {d} \sqrt {a+b x}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt {d}}-\frac {\int \frac {1}{\sqrt {b}+\sqrt {d} \sqrt {a+b x}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt {d}}\right )}{d}\right )}{8 d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 d}-\frac {7 (b c-a d) \left (\frac {(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}-\frac {3 (b c-a d) \left (\frac {\int \frac {1}{\sqrt {b}-\sqrt {d} \sqrt {a+b x}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}}}{2 \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{2 \sqrt [4]{b} d^{3/4}}\right )}{d}\right )}{8 d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 d}-\frac {7 (b c-a d) \left (\frac {(a+b x)^{3/4} \sqrt [4]{c+d x}}{d}-\frac {3 (b c-a d) \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{2 \sqrt [4]{b} d^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{\frac {d (a+b x)}{b}-\frac {a d}{b}+c}}\right )}{2 \sqrt [4]{b} d^{3/4}}\right )}{d}\right )}{8 d}\) |
((a + b*x)^(7/4)*(c + d*x)^(1/4))/(2*d) - (7*(b*c - a*d)*(((a + b*x)^(3/4) *(c + d*x)^(1/4))/d - (3*(b*c - a*d)*(-1/2*ArcTan[(d^(1/4)*(a + b*x)^(1/4) )/(b^(1/4)*(c - (a*d)/b + (d*(a + b*x))/b)^(1/4))]/(b^(1/4)*d^(3/4)) + Arc Tanh[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c - (a*d)/b + (d*(a + b*x))/b)^(1 /4))]/(2*b^(1/4)*d^(3/4))))/d))/(8*d)
3.18.5.3.1 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_)^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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 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)] && IntegersQ[m, p + (m + 1)/n]
\[\int \frac {\left (b x +a \right )^{\frac {7}{4}}}{\left (d x +c \right )^{\frac {3}{4}}}d x\]
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 1215, normalized size of antiderivative = 7.28 \[ \int \frac {(a+b x)^{7/4}}{(c+d x)^{3/4}} \, dx=\text {Too large to display} \]
1/32*(21*d^2*((b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c ^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8* a^7*b*c*d^7 + a^8*d^8)/(b*d^11))^(1/4)*log(21*((b^2*c^2 - 2*a*b*c*d + a^2* d^2)*(b*x + a)^(3/4)*(d*x + c)^(1/4) + (b*d^3*x + a*d^3)*((b^8*c^8 - 8*a*b ^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(b*d^11 ))^(1/4))/(b*x + a)) - 21*d^2*((b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d ^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6 *b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(b*d^11))^(1/4)*log(21*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(3/4)*(d*x + c)^(1/4) - (b*d^3*x + a*d^3)* ((b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a ^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(b*d^11))^(1/4))/(b*x + a)) - 21*I*d^2*((b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b ^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(b*d^11))^(1/4) *log(21*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(3/4)*(d*x + c)^(1/4) - (I*b*d^3*x + I*a*d^3)*((b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56 *a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^ 2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(b*d^11))^(1/4))/(b*x + a)) + 21*I*d^2*(( b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*...
\[ \int \frac {(a+b x)^{7/4}}{(c+d x)^{3/4}} \, dx=\int \frac {\left (a + b x\right )^{\frac {7}{4}}}{\left (c + d x\right )^{\frac {3}{4}}}\, dx \]
\[ \int \frac {(a+b x)^{7/4}}{(c+d x)^{3/4}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {7}{4}}}{{\left (d x + c\right )}^{\frac {3}{4}}} \,d x } \]
\[ \int \frac {(a+b x)^{7/4}}{(c+d x)^{3/4}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {7}{4}}}{{\left (d x + c\right )}^{\frac {3}{4}}} \,d x } \]
Timed out. \[ \int \frac {(a+b x)^{7/4}}{(c+d x)^{3/4}} \, dx=\int \frac {{\left (a+b\,x\right )}^{7/4}}{{\left (c+d\,x\right )}^{3/4}} \,d x \]
\[ \int \frac {(a+b x)^{7/4}}{(c+d x)^{3/4}} \, dx=\left (\int \frac {\left (b x +a \right )^{\frac {3}{4}}}{\left (d x +c \right )^{\frac {3}{4}}}d x \right ) a +\left (\int \frac {\left (b x +a \right )^{\frac {3}{4}} x}{\left (d x +c \right )^{\frac {3}{4}}}d x \right ) b \]