Integrand size = 29, antiderivative size = 356 \[ \int \frac {(a+b \tan (e+f x))^{7/2}}{(c+d \tan (e+f x))^{3/2}} \, dx=-\frac {i (a-i b)^{7/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(c-i d)^{3/2} f}+\frac {i (a+i b)^{7/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(c+i d)^{3/2} f}-\frac {b^{5/2} (3 b c-7 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{d^{5/2} f}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{3/2}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}-\frac {b \left (2 a d (2 b c-a d)-b^2 \left (3 c^2+d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d^2 \left (c^2+d^2\right ) f} \] Output:
-I*(a-I*b)^(7/2)*arctanh((c-I*d)^(1/2)*(a+b*tan(f*x+e))^(1/2)/(a-I*b)^(1/2 )/(c+d*tan(f*x+e))^(1/2))/(c-I*d)^(3/2)/f+I*(a+I*b)^(7/2)*arctanh((c+I*d)^ (1/2)*(a+b*tan(f*x+e))^(1/2)/(a+I*b)^(1/2)/(c+d*tan(f*x+e))^(1/2))/(c+I*d) ^(3/2)/f-b^(5/2)*(-7*a*d+3*b*c)*arctanh(d^(1/2)*(a+b*tan(f*x+e))^(1/2)/b^( 1/2)/(c+d*tan(f*x+e))^(1/2))/d^(5/2)/f-2*(-a*d+b*c)^2*(a+b*tan(f*x+e))^(3/ 2)/d/(c^2+d^2)/f/(c+d*tan(f*x+e))^(1/2)-b*(2*a*d*(-a*d+2*b*c)-b^2*(3*c^2+d ^2))*(a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(1/2)/d^2/(c^2+d^2)/f
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 6.63 (sec) , antiderivative size = 1877, normalized size of antiderivative = 5.27 \[ \int \frac {(a+b \tan (e+f x))^{7/2}}{(c+d \tan (e+f x))^{3/2}} \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*Tan[e + f*x])^(7/2)/(c + d*Tan[e + f*x])^(3/2),x]
Output:
((-1/2*I)*(-a - I*b)*((-2*b*(b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)) )^(3/2)*Hypergeometric2F1[3/2, 5/2, 7/2, -((b*d*(a + b*Tan[e + f*x]))/((b* c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))]*(a + b*Tan[e + f*x ])^(5/2)*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(5*(b*c - a*d)*Sqrt[c + d*Tan[e + f*x]]) - (-a - I*b)*(-((-a - I*b)*((-2*Sqrt[a + I*b]*ArcTanh[ (Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((-c - I*d)*Sqrt[c + I*d]) + (2*Sqrt[a + b*Tan[e + f*x]])/((-c - I*d)*Sqrt[c + d*Tan[e + f*x]]))) - (2*(b*c - a*d)*(b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))^(3/2)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)) ^2*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)]*(-1 - (b*d*(a + b*Tan[e + f* x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))*(-((b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d ))*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a *b*d)/(b*c - a*d)))))) - (Sqrt[b]*Sqrt[d]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b* c - a*d)])]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a *d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))])))/(b*d^2*Sqrt[a + b*Tan [e + f*x]]*Sqrt[c + d*Tan[e + f*x]]*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/(( b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))]))))/f - ((I/2)...
Time = 2.47 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {3042, 4048, 27, 3042, 4130, 27, 3042, 4138, 2348, 2009}
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.
| \(\displaystyle \int \frac {(a+b \tan (e+f x))^{7/2}}{(c+d \tan (e+f x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (e+f x))^{7/2}}{(c+d \tan (e+f x))^{3/2}}dx\) |
| \(\Big \downarrow \) 4048 |
| \(\displaystyle \frac {2 \int \frac {\sqrt {a+b \tan (e+f x)} \left (c d a^3+5 b d^2 a^2-7 b^2 c d a+3 b^3 c^2-b \left (2 a d (2 b c-a d)-b^2 \left (3 c^2+d^2\right )\right ) \tan ^2(e+f x)+d \left (-d a^3+3 b c a^2+3 b^2 d a-b^3 c\right ) \tan (e+f x)\right )}{2 \sqrt {c+d \tan (e+f x)}}dx}{d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{3/2}}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {a+b \tan (e+f x)} \left (c d a^3+5 b d^2 a^2-7 b^2 c d a+3 b^3 c^2-b \left (2 a d (2 b c-a d)-b^2 \left (3 c^2+d^2\right )\right ) \tan ^2(e+f x)+d \left (-d a^3+3 b c a^2+3 b^2 d a-b^3 c\right ) \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}}dx}{d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{3/2}}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sqrt {a+b \tan (e+f x)} \left (c d a^3+5 b d^2 a^2-7 b^2 c d a+3 b^3 c^2-b \left (2 a d (2 b c-a d)-b^2 \left (3 c^2+d^2\right )\right ) \tan (e+f x)^2+d \left (-d a^3+3 b c a^2+3 b^2 d a-b^3 c\right ) \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}}dx}{d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{3/2}}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\frac {\int \frac {2 c d^2 a^4+8 b d^3 a^3-12 b^2 c d^2 a^2+b^3 d \left (7 c^2-d^2\right ) a-b^3 (3 b c-7 a d) \left (c^2+d^2\right ) \tan ^2(e+f x)-b^4 c \left (3 c^2+d^2\right )+2 d^2 \left (-d a^4+4 b c a^3+6 b^2 d a^2-4 b^3 c a-b^4 d\right ) \tan (e+f x)}{2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{d}-\frac {b \left (2 a d (2 b c-a d)-b^2 \left (3 c^2+d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d f}}{d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{3/2}}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {2 c d^2 a^4+8 b d^3 a^3-12 b^2 c d^2 a^2+b^3 d \left (7 c^2-d^2\right ) a-b^3 (3 b c-7 a d) \left (c^2+d^2\right ) \tan ^2(e+f x)-b^4 c \left (3 c^2+d^2\right )+2 d^2 \left (-d a^4+4 b c a^3+6 b^2 d a^2-4 b^3 c a-b^4 d\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{2 d}-\frac {b \left (2 a d (2 b c-a d)-b^2 \left (3 c^2+d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d f}}{d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{3/2}}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {2 c d^2 a^4+8 b d^3 a^3-12 b^2 c d^2 a^2+b^3 d \left (7 c^2-d^2\right ) a-b^3 (3 b c-7 a d) \left (c^2+d^2\right ) \tan (e+f x)^2-b^4 c \left (3 c^2+d^2\right )+2 d^2 \left (-d a^4+4 b c a^3+6 b^2 d a^2-4 b^3 c a-b^4 d\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{2 d}-\frac {b \left (2 a d (2 b c-a d)-b^2 \left (3 c^2+d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d f}}{d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{3/2}}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 4138 |
| \(\displaystyle \frac {\frac {\int \frac {2 c d^2 a^4+8 b d^3 a^3-12 b^2 c d^2 a^2+b^3 d \left (7 c^2-d^2\right ) a-b^3 (3 b c-7 a d) \left (c^2+d^2\right ) \tan ^2(e+f x)-b^4 c \left (3 c^2+d^2\right )+2 d^2 \left (-d a^4+4 b c a^3+6 b^2 d a^2-4 b^3 c a-b^4 d\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (\tan ^2(e+f x)+1\right )}d\tan (e+f x)}{2 d f}-\frac {b \left (2 a d (2 b c-a d)-b^2 \left (3 c^2+d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d f}}{d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{3/2}}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
| \(\Big \downarrow \) 2348 |
| \(\displaystyle -\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{3/2}}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {-\frac {b \left (2 a d (2 b c-a d)-b^2 \left (3 c^2+d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d f}+\frac {\int \left (-\frac {(3 b c-7 a d) \left (c^2+d^2\right ) b^3}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {2 d^3 a^4-8 b c d^2 a^3-12 b^2 d^3 a^2+8 b^3 c d^2 a+2 b^4 d^3+i \left (2 c d^2 a^4+8 b d^3 a^3-12 b^2 c d^2 a^2-8 b^3 d^3 a+2 b^4 c d^2\right )}{2 (i-\tan (e+f x)) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {-2 d^3 a^4+8 b c d^2 a^3+12 b^2 d^3 a^2-8 b^3 c d^2 a-2 b^4 d^3+i \left (2 c d^2 a^4+8 b d^3 a^3-12 b^2 c d^2 a^2-8 b^3 d^3 a+2 b^4 c d^2\right )}{2 (\tan (e+f x)+i) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}\right )d\tan (e+f x)}{2 d f}}{d \left (c^2+d^2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{3/2}}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {-\frac {b \left (2 a d (2 b c-a d)-b^2 \left (3 c^2+d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d f}+\frac {-\frac {2 b^{5/2} \left (c^2+d^2\right ) (3 b c-7 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {d}}-\frac {2 d^2 (a-i b)^{7/2} (-d+i c) \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {c-i d}}+\frac {2 d^2 (a+i b)^{7/2} (d+i c) \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {c+i d}}}{2 d f}}{d \left (c^2+d^2\right )}\) |
Input:
Int[(a + b*Tan[e + f*x])^(7/2)/(c + d*Tan[e + f*x])^(3/2),x]
Output:
(-2*(b*c - a*d)^2*(a + b*Tan[e + f*x])^(3/2))/(d*(c^2 + d^2)*f*Sqrt[c + d* Tan[e + f*x]]) + (((-2*(a - I*b)^(7/2)*(I*c - d)*d^2*ArcTanh[(Sqrt[c - I*d ]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/Sqr t[c - I*d] + (2*(a + I*b)^(7/2)*d^2*(I*c + d)*ArcTanh[(Sqrt[c + I*d]*Sqrt[ a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[c + I *d] - (2*b^(5/2)*(3*b*c - 7*a*d)*(c^2 + d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*T an[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[d])/(2*d*f) - (b*( 2*a*d*(2*b*c - a*d) - b^2*(3*c^2 + d^2))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(d*f))/(d*(c^2 + d^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Px_)*((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_. )*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(c + d*x)^m*(e + f*x)^ n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[P x, x] && (IntegerQ[p] || (IntegerQ[2*p] && IntegerQ[m] && ILtQ[n, 0])) && !(IGtQ[m, 0] && IGtQ[n, 0])
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Simp[1 /(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c *(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d) *Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*( n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 2] && LtQ [n, -1] && IntegerQ[2*m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_. ) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d*Tan[ e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C *(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f*x] - (C* m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[ c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, S imp[ff/f Subst[Int[(a + b*ff*x)^m*(c + d*ff*x)^n*((A + B*ff*x + C*ff^2*x^ 2)/(1 + ff^2*x^2)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f , A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Timed out.
\[\int \frac {\left (a +b \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{\left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}d x\]
Input:
int((a+b*tan(f*x+e))^(7/2)/(c+d*tan(f*x+e))^(3/2),x)
Output:
int((a+b*tan(f*x+e))^(7/2)/(c+d*tan(f*x+e))^(3/2),x)
Leaf count of result is larger than twice the leaf count of optimal. 27455 vs. \(2 (290) = 580\).
Time = 77.75 (sec) , antiderivative size = 54970, normalized size of antiderivative = 154.41 \[ \int \frac {(a+b \tan (e+f x))^{7/2}}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate((a+b*tan(f*x+e))^(7/2)/(c+d*tan(f*x+e))^(3/2),x, algorithm="fric as")
Output:
Too large to include
Timed out. \[ \int \frac {(a+b \tan (e+f x))^{7/2}}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((a+b*tan(f*x+e))**(7/2)/(c+d*tan(f*x+e))**(3/2),x)
Output:
Timed out
\[ \int \frac {(a+b \tan (e+f x))^{7/2}}{(c+d \tan (e+f x))^{3/2}} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*tan(f*x+e))^(7/2)/(c+d*tan(f*x+e))^(3/2),x, algorithm="maxi ma")
Output:
integrate((b*tan(f*x + e) + a)^(7/2)/(d*tan(f*x + e) + c)^(3/2), x)
\[ \int \frac {(a+b \tan (e+f x))^{7/2}}{(c+d \tan (e+f x))^{3/2}} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*tan(f*x+e))^(7/2)/(c+d*tan(f*x+e))^(3/2),x, algorithm="giac ")
Output:
integrate((b*tan(f*x + e) + a)^(7/2)/(d*tan(f*x + e) + c)^(3/2), x)
Timed out. \[ \int \frac {(a+b \tan (e+f x))^{7/2}}{(c+d \tan (e+f x))^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{7/2}}{{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}} \,d x \] Input:
int((a + b*tan(e + f*x))^(7/2)/(c + d*tan(e + f*x))^(3/2),x)
Output:
int((a + b*tan(e + f*x))^(7/2)/(c + d*tan(e + f*x))^(3/2), x)
\[ \int \frac {(a+b \tan (e+f x))^{7/2}}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {too large to display} \] Input:
int((a+b*tan(f*x+e))^(7/2)/(c+d*tan(f*x+e))^(3/2),x)
Output:
( - 6*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*a**2*b**2 + int((s qrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)**3)/(tan(e + f*x)**2*d**2 + 2*tan(e + f*x)*c*d + c**2),x)*tan(e + f*x)*a*b**3*d**2*f - int((sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)**3)/( tan(e + f*x)**2*d**2 + 2*tan(e + f*x)*c*d + c**2),x)*tan(e + f*x)*b**4*c*d *f + int((sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)** 3)/(tan(e + f*x)**2*d**2 + 2*tan(e + f*x)*c*d + c**2),x)*a*b**3*c*d*f - in t((sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)**3)/(tan (e + f*x)**2*d**2 + 2*tan(e + f*x)*c*d + c**2),x)*b**4*c**2*f + 3*int((sqr t(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)**2)/(tan(e + f *x)**2*d**2 + 2*tan(e + f*x)*c*d + c**2),x)*tan(e + f*x)*a**2*b**2*d**2*f - 3*int((sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)**2 )/(tan(e + f*x)**2*d**2 + 2*tan(e + f*x)*c*d + c**2),x)*tan(e + f*x)*a*b** 3*c*d*f + 3*int((sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)**2)/(tan(e + f*x)**2*d**2 + 2*tan(e + f*x)*c*d + c**2),x)*a**2*b**2* c*d*f - 3*int((sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f *x)**2)/(tan(e + f*x)**2*d**2 + 2*tan(e + f*x)*c*d + c**2),x)*a*b**3*c**2* f + 3*int((sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)) /(tan(e + f*x)**3*b*d**2 + tan(e + f*x)**2*a*d**2 + 2*tan(e + f*x)**2*b*c* d + 2*tan(e + f*x)*a*c*d + tan(e + f*x)*b*c**2 + a*c**2),x)*tan(e + f*x...