Integrand size = 33, antiderivative size = 187 \[ \int \tan ^m(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\frac {(A+i B) \operatorname {AppellF1}\left (1+m,-\frac {1}{2},1,2+m,-\frac {b \tan (c+d x)}{a},-i \tan (c+d x)\right ) \tan ^{1+m}(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d (1+m) \sqrt {1+\frac {b \tan (c+d x)}{a}}}+\frac {(A-i B) \operatorname {AppellF1}\left (1+m,-\frac {1}{2},1,2+m,-\frac {b \tan (c+d x)}{a},i \tan (c+d x)\right ) \tan ^{1+m}(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d (1+m) \sqrt {1+\frac {b \tan (c+d x)}{a}}} \]
1/2*(A+I*B)*AppellF1(1+m,1,-1/2,2+m,-I*tan(d*x+c),-b*tan(d*x+c)/a)*(a+b*ta n(d*x+c))^(1/2)*tan(d*x+c)^(1+m)/d/(1+m)/(1+b*tan(d*x+c)/a)^(1/2)+1/2*(A-I *B)*AppellF1(1+m,1,-1/2,2+m,I*tan(d*x+c),-b*tan(d*x+c)/a)*(a+b*tan(d*x+c)) ^(1/2)*tan(d*x+c)^(1+m)/d/(1+m)/(1+b*tan(d*x+c)/a)^(1/2)
\[ \int \tan ^m(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\int \tan ^m(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx \]
Time = 0.61 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3042, 4086, 3042, 4085, 152, 150}
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.
\(\displaystyle \int \tan ^m(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \tan (c+d x)^m \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x))dx\) |
\(\Big \downarrow \) 4086 |
\(\displaystyle \frac {1}{2} (A+i B) \int (1-i \tan (c+d x)) \tan ^m(c+d x) \sqrt {a+b \tan (c+d x)}dx+\frac {1}{2} (A-i B) \int (i \tan (c+d x)+1) \tan ^m(c+d x) \sqrt {a+b \tan (c+d x)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} (A+i B) \int (1-i \tan (c+d x)) \tan (c+d x)^m \sqrt {a+b \tan (c+d x)}dx+\frac {1}{2} (A-i B) \int (i \tan (c+d x)+1) \tan (c+d x)^m \sqrt {a+b \tan (c+d x)}dx\) |
\(\Big \downarrow \) 4085 |
\(\displaystyle \frac {(A-i B) \int \frac {\tan ^m(c+d x) \sqrt {a+b \tan (c+d x)}}{1-i \tan (c+d x)}d\tan (c+d x)}{2 d}+\frac {(A+i B) \int \frac {\tan ^m(c+d x) \sqrt {a+b \tan (c+d x)}}{i \tan (c+d x)+1}d\tan (c+d x)}{2 d}\) |
\(\Big \downarrow \) 152 |
\(\displaystyle \frac {(A-i B) \sqrt {a+b \tan (c+d x)} \int \frac {\tan ^m(c+d x) \sqrt {\frac {b \tan (c+d x)}{a}+1}}{1-i \tan (c+d x)}d\tan (c+d x)}{2 d \sqrt {\frac {b \tan (c+d x)}{a}+1}}+\frac {(A+i B) \sqrt {a+b \tan (c+d x)} \int \frac {\tan ^m(c+d x) \sqrt {\frac {b \tan (c+d x)}{a}+1}}{i \tan (c+d x)+1}d\tan (c+d x)}{2 d \sqrt {\frac {b \tan (c+d x)}{a}+1}}\) |
\(\Big \downarrow \) 150 |
\(\displaystyle \frac {(A+i B) \tan ^{m+1}(c+d x) \sqrt {a+b \tan (c+d x)} \operatorname {AppellF1}\left (m+1,-\frac {1}{2},1,m+2,-\frac {b \tan (c+d x)}{a},-i \tan (c+d x)\right )}{2 d (m+1) \sqrt {\frac {b \tan (c+d x)}{a}+1}}+\frac {(A-i B) \tan ^{m+1}(c+d x) \sqrt {a+b \tan (c+d x)} \operatorname {AppellF1}\left (m+1,-\frac {1}{2},1,m+2,-\frac {b \tan (c+d x)}{a},i \tan (c+d x)\right )}{2 d (m+1) \sqrt {\frac {b \tan (c+d x)}{a}+1}}\) |
((A + I*B)*AppellF1[1 + m, -1/2, 1, 2 + m, -((b*Tan[c + d*x])/a), (-I)*Tan [c + d*x]]*Tan[c + d*x]^(1 + m)*Sqrt[a + b*Tan[c + d*x]])/(2*d*(1 + m)*Sqr t[1 + (b*Tan[c + d*x])/a]) + ((A - I*B)*AppellF1[1 + m, -1/2, 1, 2 + m, -( (b*Tan[c + d*x])/a), I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*Sqrt[a + b*Tan[c + d*x]])/(2*d*(1 + m)*Sqrt[1 + (b*Tan[c + d*x])/a])
3.5.89.3.1 Defintions of rubi rules used
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ ] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 , (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] && !In tegerQ[m] && !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ ] :> Simp[c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n]) Int[(b*x)^m*(1 + d*(x/c))^n*(e + f*x)^p, x], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !GtQ[c, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[A^2/f Subst[Int[(a + b*x)^m*((c + d*x)^n/(A - B*x)), x], x, Tan[e + f*x ]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && !IntegerQ[m] && !IntegerQ[n] && !IntegersQ[2*m, 2*n ] && EqQ[A^2 + B^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(A + I*B)/2 Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(1 - I*T an[e + f*x]), x], x] + Simp[(A - I*B)/2 Int[(a + b*Tan[e + f*x])^m*(c + d *Tan[e + f*x])^n*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A , B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && !IntegerQ[m] & & !IntegerQ[n] && !IntegersQ[2*m, 2*n] && NeQ[A^2 + B^2, 0]
\[\int \tan \left (d x +c \right )^{m} \sqrt {a +b \tan \left (d x +c \right )}\, \left (A +B \tan \left (d x +c \right )\right )d x\]
\[ \int \tan ^m(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} \sqrt {b \tan \left (d x + c\right ) + a} \tan \left (d x + c\right )^{m} \,d x } \]
\[ \int \tan ^m(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\int \left (A + B \tan {\left (c + d x \right )}\right ) \sqrt {a + b \tan {\left (c + d x \right )}} \tan ^{m}{\left (c + d x \right )}\, dx \]
\[ \int \tan ^m(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} \sqrt {b \tan \left (d x + c\right ) + a} \tan \left (d x + c\right )^{m} \,d x } \]
\[ \int \tan ^m(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} \sqrt {b \tan \left (d x + c\right ) + a} \tan \left (d x + c\right )^{m} \,d x } \]
Timed out. \[ \int \tan ^m(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\int {\mathrm {tan}\left (c+d\,x\right )}^m\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )} \,d x \]