Integrand size = 49, antiderivative size = 697 \[ \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=-\frac {\sqrt {a-i b} (i A+B-i C) (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {\sqrt {a+i b} (i A-B-i C) (c+i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {\left (5 a^4 C d^4-4 a^3 b d^3 (5 c C+2 B d)+2 a^2 b^2 d^2 \left (15 c^2 C+20 B c d+8 (A-C) d^2\right )-4 a b^3 d \left (5 c^3 C+30 B c^2 d+40 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-40 B c^3 d-240 c^2 (A-C) d^2+320 B c d^3+128 (A-C) d^4\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{64 b^{7/2} d^{3/2} f}+\frac {\left (64 b^2 d^2 (A b c+a B c-b c C+a A d-b B d-a C d)+(b c-a d) \left (48 b (A b+a B-b C) d^2-5 (b c-a d) (b c C-8 b B d-a C d)\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{64 b^3 d f}+\frac {\left (48 b (A b+a B-b C) d^2-5 (b c-a d) (b c C-8 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b^2 d f}-\frac {(b c C-8 b B d-a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 b d f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f} \]
-1/64*(5*a^4*C*d^4-4*a^3*b*d^3*(2*B*d+5*C*c)+2*a^2*b^2*d^2*(15*c^2*C+20*B* c*d+8*(A-C)*d^2)-4*a*b^3*d*(5*c^3*C+30*B*c^2*d+40*c*(A-C)*d^2-16*B*d^3)+b^ 4*(5*c^4*C-40*B*c^3*d-240*c^2*(A-C)*d^2+320*B*c*d^3+128*(A-C)*d^4))*arctan h(d^(1/2)*(a+b*tan(f*x+e))^(1/2)/b^(1/2)/(c+d*tan(f*x+e))^(1/2))/b^(7/2)/d ^(3/2)/f-(I*A+B-I*C)*(c-I*d)^(5/2)*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))*(a-I*b)^(1/2)/f+(I*A-B-I*C)*(c +I*d)^(5/2)*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))*(a+I*b)^(1/2)/f+1/64*(64*b^2*d^2*(A*a*d+A*b*c+B*a*c-B *b*d-C*a*d-C*b*c)+(-a*d+b*c)*(48*b*(A*b+B*a-C*b)*d^2-5*(-a*d+b*c)*(-8*B*b* d-C*a*d+C*b*c)))*(a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(1/2)/b^3/d/f+1/9 6*(48*b*(A*b+B*a-C*b)*d^2-5*(-a*d+b*c)*(-8*B*b*d-C*a*d+C*b*c))*(a+b*tan(f* x+e))^(1/2)*(c+d*tan(f*x+e))^(3/2)/b^2/d/f-1/24*(-8*B*b*d-C*a*d+C*b*c)*(a+ b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(5/2)/b/d/f+1/4*C*(a+b*tan(f*x+e))^(1 /2)*(c+d*tan(f*x+e))^(7/2)/d/f
Time = 9.88 (sec) , antiderivative size = 1261, normalized size of antiderivative = 1.81 \[ \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}+\frac {\frac {(-b c C+8 b B d+a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{6 b f}+\frac {\frac {\left (48 b (A b+a B-b C) d^2-5 (b c-a d) (b c C-8 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{8 b f}+\frac {\frac {\left (24 b^2 d^2 (A b c+a B c-b c C+a A d-b B d-a C d)-\frac {3}{8} (-b c+a d) \left (48 b (A b+a B-b C) d^2-5 (b c-a d) (b c C-8 b B d-a C d)\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b f}+\frac {-\frac {24 b^3 d \left (\sqrt {-b^2} \left (b (A-C) d \left (3 c^2-d^2\right )+b B \left (c^3-3 c d^2\right )-a \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )\right )-b \left (A \left (b c^3+3 a c^2 d-3 b c d^2-a d^3\right )-b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )+a \left (B c^3-3 c^2 C d-3 B c d^2+C d^3\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {-c+\frac {\sqrt {-b^2} d}{b}} \sqrt {a+b \tan (e+f x)}}{\sqrt {-a+\sqrt {-b^2}} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {-a+\sqrt {-b^2}} \sqrt {-c+\frac {\sqrt {-b^2} d}{b}}}-\frac {24 b^3 d \left (\sqrt {-b^2} \left (b (A-C) d \left (3 c^2-d^2\right )+b B \left (c^3-3 c d^2\right )-a \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )\right )+b \left (A \left (b c^3+3 a c^2 d-3 b c d^2-a d^3\right )-b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )+a \left (B c^3-3 c^2 C d-3 B c d^2+C d^3\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {c+\frac {\sqrt {-b^2} d}{b}} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+\sqrt {-b^2}} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+\sqrt {-b^2}} \sqrt {c+\frac {\sqrt {-b^2} d}{b}}}-\frac {3 \sqrt {b} \sqrt {c-\frac {a d}{b}} \sqrt {\frac {1}{\frac {c}{c-\frac {a d}{b}}-\frac {a d}{b \left (c-\frac {a d}{b}\right )}}} \sqrt {\frac {c}{c-\frac {a d}{b}}-\frac {a d}{b \left (c-\frac {a d}{b}\right )}} \left (5 a^4 C d^4-4 a^3 b d^3 (5 c C+2 B d)+2 a^2 b^2 d^2 \left (15 c^2 C+20 B c d+8 (A-C) d^2\right )-4 a b^3 d \left (5 c^3 C+30 B c^2 d+40 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-40 B c^3 d-240 c^2 (A-C) d^2+320 B c d^3+128 (A-C) d^4\right )\right ) \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c-\frac {a d}{b}} \sqrt {\frac {c}{c-\frac {a d}{b}}-\frac {a d}{b \left (c-\frac {a d}{b}\right )}}}\right ) \sqrt {\frac {c+d \tan (e+f x)}{c-\frac {a d}{b}}}}{8 \sqrt {d} \sqrt {c+d \tan (e+f x)}}}{b^2 f}}{2 b}}{3 b}}{4 d} \]
Integrate[Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]
(C*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(7/2))/(4*d*f) + (((-(b*c *C) + 8*b*B*d + a*C*d)*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2) )/(6*b*f) + (((48*b*(A*b + a*B - b*C)*d^2 - 5*(b*c - a*d)*(b*c*C - 8*b*B*d - a*C*d))*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(8*b*f) + (((24*b^2*d^2*(A*b*c + a*B*c - b*c*C + a*A*d - b*B*d - a*C*d) - (3*(-(b*c) + a*d)*(48*b*(A*b + a*B - b*C)*d^2 - 5*(b*c - a*d)*(b*c*C - 8*b*B*d - a*C *d)))/8)*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(b*f) + ((-24* b^3*d*(Sqrt[-b^2]*(b*(A - C)*d*(3*c^2 - d^2) + b*B*(c^3 - 3*c*d^2) - a*(A* c^3 - c^3*C - 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 + B*d^3)) - b*(A*(b*c^3 + 3*a*c^2*d - 3*b*c*d^2 - a*d^3) - b*(c^3*C + 3*B*c^2*d - 3*c*C*d^2 - B*d^3) + a*(B*c^3 - 3*c^2*C*d - 3*B*c*d^2 + C*d^3)))*ArcTanh[(Sqrt[-c + (Sqrt[-b ^2]*d)/b]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + Sqrt[-b^2]]*Sqrt[c + d*Tan[ e + f*x]])])/(Sqrt[-a + Sqrt[-b^2]]*Sqrt[-c + (Sqrt[-b^2]*d)/b]) - (24*b^3 *d*(Sqrt[-b^2]*(b*(A - C)*d*(3*c^2 - d^2) + b*B*(c^3 - 3*c*d^2) - a*(A*c^3 - c^3*C - 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 + B*d^3)) + b*(A*(b*c^3 + 3*a *c^2*d - 3*b*c*d^2 - a*d^3) - b*(c^3*C + 3*B*c^2*d - 3*c*C*d^2 - B*d^3) + a*(B*c^3 - 3*c^2*C*d - 3*B*c*d^2 + C*d^3)))*ArcTanh[(Sqrt[c + (Sqrt[-b^2]* d)/b]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + Sqrt[-b^2]]*Sqrt[c + d*Tan[e + f *x]])])/(Sqrt[a + Sqrt[-b^2]]*Sqrt[c + (Sqrt[-b^2]*d)/b]) - (3*Sqrt[b]*Sqr t[c - (a*d)/b]*Sqrt[(c/(c - (a*d)/b) - (a*d)/(b*(c - (a*d)/b)))^(-1)]*S...
Time = 5.85 (sec) , antiderivative size = 717, normalized size of antiderivative = 1.03, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.327, Rules used = {3042, 4130, 27, 3042, 4130, 27, 3042, 4130, 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.
| \(\displaystyle \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan (e+f x)^2\right )dx\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\int -\frac {(c+d \tan (e+f x))^{5/2} \left ((b c C-a d C-8 b B d) \tan ^2(e+f x)-8 (A b-C b+a B) d \tan (e+f x)+b c C-a (8 A-7 C) d\right )}{2 \sqrt {a+b \tan (e+f x)}}dx}{4 d}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}-\frac {\int \frac {(c+d \tan (e+f x))^{5/2} \left ((b c C-a d C-8 b B d) \tan ^2(e+f x)-8 (A b-C b+a B) d \tan (e+f x)+b c C-a (8 A-7 C) d\right )}{\sqrt {a+b \tan (e+f x)}}dx}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}-\frac {\int \frac {(c+d \tan (e+f x))^{5/2} \left ((b c C-a d C-8 b B d) \tan (e+f x)^2-8 (A b-C b+a B) d \tan (e+f x)+b c C-a (8 A-7 C) d\right )}{\sqrt {a+b \tan (e+f x)}}dx}{8 d}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}-\frac {\frac {\int \frac {(c+d \tan (e+f x))^{3/2} \left (c (5 c C+8 B d) b^2-2 a d (24 A c-19 C c-20 B d) b-48 d (A b c+a B c-b C c+a A d-b B d-a C d) \tan (e+f x) b+5 a^2 C d^2-\left (48 b (A b-C b+a B) d^2-5 (b c-a d) (b c C-a d C-8 b B d)\right ) \tan ^2(e+f x)\right )}{2 \sqrt {a+b \tan (e+f x)}}dx}{3 b}+\frac {(-a C d-8 b B d+b c C) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 b f}}{8 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}-\frac {\frac {\int \frac {(c+d \tan (e+f x))^{3/2} \left (c (5 c C+8 B d) b^2-2 a d (24 A c-19 C c-20 B d) b-48 d (A b c+a B c-b C c+a A d-b B d-a C d) \tan (e+f x) b+5 a^2 C d^2-\left (48 b (A b-C b+a B) d^2-5 (b c-a d) (b c C-a d C-8 b B d)\right ) \tan ^2(e+f x)\right )}{\sqrt {a+b \tan (e+f x)}}dx}{6 b}+\frac {(-a C d-8 b B d+b c C) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 b f}}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}-\frac {\frac {\int \frac {(c+d \tan (e+f x))^{3/2} \left (c (5 c C+8 B d) b^2-2 a d (24 A c-19 C c-20 B d) b-48 d (A b c+a B c-b C c+a A d-b B d-a C d) \tan (e+f x) b+5 a^2 C d^2-\left (48 b (A b-C b+a B) d^2-5 (b c-a d) (b c C-a d C-8 b B d)\right ) \tan (e+f x)^2\right )}{\sqrt {a+b \tan (e+f x)}}dx}{6 b}+\frac {(-a C d-8 b B d+b c C) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 b f}}{8 d}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}-\frac {\frac {\frac {\int -\frac {3 \sqrt {c+d \tan (e+f x)} \left (-c \left (5 C c^2+24 B d c+16 (A-C) d^2\right ) b^3+a d \left (64 A c^2-49 C c^2-96 B d c-48 A d^2+48 C d^2\right ) b^2+64 d \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (C c^2+2 B d c-C d^2\right )\right ) \tan (e+f x) b^2-a^2 d^2 (15 c C+8 B d) b+5 a^3 C d^3+\left (64 b^2 (A b c+a B c-b C c+a A d-b B d-a C d) d^2+(b c-a d) \left (48 b (A b-C b+a B) d^2-5 (b c-a d) (b c C-a d C-8 b B d)\right )\right ) \tan ^2(e+f x)\right )}{2 \sqrt {a+b \tan (e+f x)}}dx}{2 b}-\frac {\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (48 b d^2 (a B+A b-b C)-5 (b c-a d) (-a C d-8 b B d+b c C)\right )}{2 b f}}{6 b}+\frac {(-a C d-8 b B d+b c C) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 b f}}{8 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}-\frac {\frac {-\frac {3 \int \frac {\sqrt {c+d \tan (e+f x)} \left (-c \left (5 C c^2+24 B d c+16 (A-C) d^2\right ) b^3+a d \left (64 A c^2-49 C c^2-96 B d c-48 A d^2+48 C d^2\right ) b^2+64 d \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (C c^2+2 B d c-C d^2\right )\right ) \tan (e+f x) b^2-a^2 d^2 (15 c C+8 B d) b+5 a^3 C d^3+\left (64 b^2 (A b c+a B c-b C c+a A d-b B d-a C d) d^2+(b c-a d) \left (48 b (A b-C b+a B) d^2-5 (b c-a d) (b c C-a d C-8 b B d)\right )\right ) \tan ^2(e+f x)\right )}{\sqrt {a+b \tan (e+f x)}}dx}{4 b}-\frac {\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (48 b d^2 (a B+A b-b C)-5 (b c-a d) (-a C d-8 b B d+b c C)\right )}{2 b f}}{6 b}+\frac {(-a C d-8 b B d+b c C) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 b f}}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}-\frac {\frac {-\frac {3 \int \frac {\sqrt {c+d \tan (e+f x)} \left (-c \left (5 C c^2+24 B d c+16 (A-C) d^2\right ) b^3+a d \left (64 A c^2-49 C c^2-96 B d c-48 A d^2+48 C d^2\right ) b^2+64 d \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (C c^2+2 B d c-C d^2\right )\right ) \tan (e+f x) b^2-a^2 d^2 (15 c C+8 B d) b+5 a^3 C d^3+\left (64 b^2 (A b c+a B c-b C c+a A d-b B d-a C d) d^2+(b c-a d) \left (48 b (A b-C b+a B) d^2-5 (b c-a d) (b c C-a d C-8 b B d)\right )\right ) \tan (e+f x)^2\right )}{\sqrt {a+b \tan (e+f x)}}dx}{4 b}-\frac {\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (48 b d^2 (a B+A b-b C)-5 (b c-a d) (-a C d-8 b B d+b c C)\right )}{2 b f}}{6 b}+\frac {(-a C d-8 b B d+b c C) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 b f}}{8 d}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}-\frac {\frac {-\frac {3 \left (\frac {\int -\frac {c \left (5 C c^3+88 B d c^2+144 (A-C) d^2 c-64 B d^3\right ) b^4+4 a d \left (27 C c^3+66 B d c^2-56 C d^2 c-16 B d^3-8 A \left (4 c^3-7 c d^2\right )\right ) b^3-128 d \left (A \left (b c^3+3 a d c^2-3 b d^2 c-a d^3\right )-b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3\right )+a \left (B c^3-3 C d c^2-3 B d^2 c+C d^3\right )\right ) \tan (e+f x) b^3+2 a^2 d^2 \left (15 C c^2+20 B d c+8 (A-C) d^2\right ) b^2-4 a^3 d^3 (5 c C+2 B d) b+5 a^4 C d^4+\left (\left (5 C c^4-40 B d c^3-240 (A-C) d^2 c^2+320 B d^3 c+128 (A-C) d^4\right ) b^4-4 a d \left (5 C c^3+30 B d c^2+40 (A-C) d^2 c-16 B d^3\right ) b^3+2 a^2 d^2 \left (15 C c^2+20 B d c+8 (A-C) d^2\right ) b^2-4 a^3 d^3 (5 c C+2 B d) b+5 a^4 C d^4\right ) \tan ^2(e+f x)}{2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{b}+\frac {\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (64 b^2 d^2 (a A d+a B c-a C d+A b c-b B d-b c C)+(b c-a d) \left (48 b d^2 (a B+A b-b C)-5 (b c-a d) (-a C d-8 b B d+b c C)\right )\right )}{b f}\right )}{4 b}-\frac {\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (48 b d^2 (a B+A b-b C)-5 (b c-a d) (-a C d-8 b B d+b c C)\right )}{2 b f}}{6 b}+\frac {(-a C d-8 b B d+b c C) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 b f}}{8 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}-\frac {\frac {-\frac {3 \left (\frac {\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (64 b^2 d^2 (a A d+a B c-a C d+A b c-b B d-b c C)+(b c-a d) \left (48 b d^2 (a B+A b-b C)-5 (b c-a d) (-a C d-8 b B d+b c C)\right )\right )}{b f}-\frac {\int \frac {c \left (5 C c^3+88 B d c^2+144 (A-C) d^2 c-64 B d^3\right ) b^4+4 a d \left (27 C c^3+66 B d c^2-56 C d^2 c-16 B d^3-8 A \left (4 c^3-7 c d^2\right )\right ) b^3-128 d \left (A \left (b c^3+3 a d c^2-3 b d^2 c-a d^3\right )-b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3\right )+a \left (B c^3-3 C d c^2-3 B d^2 c+C d^3\right )\right ) \tan (e+f x) b^3+2 a^2 d^2 \left (15 C c^2+20 B d c+8 (A-C) d^2\right ) b^2-4 a^3 d^3 (5 c C+2 B d) b+5 a^4 C d^4+\left (\left (5 C c^4-40 B d c^3-240 (A-C) d^2 c^2+320 B d^3 c+128 (A-C) d^4\right ) b^4-4 a d \left (5 C c^3+30 B d c^2+40 (A-C) d^2 c-16 B d^3\right ) b^3+2 a^2 d^2 \left (15 C c^2+20 B d c+8 (A-C) d^2\right ) b^2-4 a^3 d^3 (5 c C+2 B d) b+5 a^4 C d^4\right ) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{2 b}\right )}{4 b}-\frac {\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (48 b d^2 (a B+A b-b C)-5 (b c-a d) (-a C d-8 b B d+b c C)\right )}{2 b f}}{6 b}+\frac {(-a C d-8 b B d+b c C) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 b f}}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}-\frac {\frac {-\frac {3 \left (\frac {\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (64 b^2 d^2 (a A d+a B c-a C d+A b c-b B d-b c C)+(b c-a d) \left (48 b d^2 (a B+A b-b C)-5 (b c-a d) (-a C d-8 b B d+b c C)\right )\right )}{b f}-\frac {\int \frac {c \left (5 C c^3+88 B d c^2+144 (A-C) d^2 c-64 B d^3\right ) b^4+4 a d \left (27 C c^3+66 B d c^2-56 C d^2 c-16 B d^3-8 A \left (4 c^3-7 c d^2\right )\right ) b^3-128 d \left (A \left (b c^3+3 a d c^2-3 b d^2 c-a d^3\right )-b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3\right )+a \left (B c^3-3 C d c^2-3 B d^2 c+C d^3\right )\right ) \tan (e+f x) b^3+2 a^2 d^2 \left (15 C c^2+20 B d c+8 (A-C) d^2\right ) b^2-4 a^3 d^3 (5 c C+2 B d) b+5 a^4 C d^4+\left (\left (5 C c^4-40 B d c^3-240 (A-C) d^2 c^2+320 B d^3 c+128 (A-C) d^4\right ) b^4-4 a d \left (5 C c^3+30 B d c^2+40 (A-C) d^2 c-16 B d^3\right ) b^3+2 a^2 d^2 \left (15 C c^2+20 B d c+8 (A-C) d^2\right ) b^2-4 a^3 d^3 (5 c C+2 B d) b+5 a^4 C d^4\right ) \tan (e+f x)^2}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{2 b}\right )}{4 b}-\frac {\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (48 b d^2 (a B+A b-b C)-5 (b c-a d) (-a C d-8 b B d+b c C)\right )}{2 b f}}{6 b}+\frac {(-a C d-8 b B d+b c C) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 b f}}{8 d}\) |
| \(\Big \downarrow \) 4138 |
| \(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}-\frac {\frac {-\frac {3 \left (\frac {\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (64 b^2 d^2 (a A d+a B c-a C d+A b c-b B d-b c C)+(b c-a d) \left (48 b d^2 (a B+A b-b C)-5 (b c-a d) (-a C d-8 b B d+b c C)\right )\right )}{b f}-\frac {\int \frac {c \left (5 C c^3+88 B d c^2+144 (A-C) d^2 c-64 B d^3\right ) b^4+4 a d \left (27 C c^3+66 B d c^2-56 C d^2 c-16 B d^3-8 A \left (4 c^3-7 c d^2\right )\right ) b^3-128 d \left (A \left (b c^3+3 a d c^2-3 b d^2 c-a d^3\right )-b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3\right )+a \left (B c^3-3 C d c^2-3 B d^2 c+C d^3\right )\right ) \tan (e+f x) b^3+2 a^2 d^2 \left (15 C c^2+20 B d c+8 (A-C) d^2\right ) b^2-4 a^3 d^3 (5 c C+2 B d) b+5 a^4 C d^4+\left (\left (5 C c^4-40 B d c^3-240 (A-C) d^2 c^2+320 B d^3 c+128 (A-C) d^4\right ) b^4-4 a d \left (5 C c^3+30 B d c^2+40 (A-C) d^2 c-16 B d^3\right ) b^3+2 a^2 d^2 \left (15 C c^2+20 B d c+8 (A-C) d^2\right ) b^2-4 a^3 d^3 (5 c C+2 B d) b+5 a^4 C d^4\right ) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (\tan ^2(e+f x)+1\right )}d\tan (e+f x)}{2 b f}\right )}{4 b}-\frac {\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (48 b d^2 (a B+A b-b C)-5 (b c-a d) (-a C d-8 b B d+b c C)\right )}{2 b f}}{6 b}+\frac {(-a C d-8 b B d+b c C) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 b f}}{8 d}\) |
| \(\Big \downarrow \) 2348 |
| \(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}-\frac {\frac {(b c C-a d C-8 b B d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 b f}+\frac {-\frac {\left (48 b (A b-C b+a B) d^2-5 (b c-a d) (b c C-a d C-8 b B d)\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}-\frac {3 \left (\frac {\left (64 b^2 (A b c+a B c-b C c+a A d-b B d-a C d) d^2+(b c-a d) \left (48 b (A b-C b+a B) d^2-5 (b c-a d) (b c C-a d C-8 b B d)\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b f}-\frac {\int \left (\frac {128 A d^4 b^4-128 C d^4 b^4+320 B c d^3 b^4-240 A c^2 d^2 b^4+240 c^2 C d^2 b^4+5 c^4 C b^4-40 B c^3 d b^4+64 a B d^4 b^3-160 a A c d^3 b^3+160 a c C d^3 b^3-120 a B c^2 d^2 b^3-20 a c^3 C d b^3+16 a^2 A d^4 b^2-16 a^2 C d^4 b^2+40 a^2 B c d^3 b^2+30 a^2 c^2 C d^2 b^2-8 a^3 B d^4 b-20 a^3 c C d^3 b+5 a^4 C d^4}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {128 B d^4 b^4-384 A c d^3 b^4+384 c C d^3 b^4-384 B c^2 d^2 b^4+128 A c^3 d b^4-128 c^3 C d b^4-128 a A d^4 b^3+128 a C d^4 b^3-384 a B c d^3 b^3+384 a A c^2 d^2 b^3-384 a c^2 C d^2 b^3+128 a B c^3 d b^3+i \left (-128 A d^4 b^4+128 C d^4 b^4-384 B c d^3 b^4+384 A c^2 d^2 b^4-384 c^2 C d^2 b^4+128 B c^3 d b^4-128 a B d^4 b^3+384 a A c d^3 b^3-384 a c C d^3 b^3+384 a B c^2 d^2 b^3-128 a A c^3 d b^3+128 a c^3 C d b^3\right )}{2 (i-\tan (e+f x)) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {-128 B d^4 b^4+384 A c d^3 b^4-384 c C d^3 b^4+384 B c^2 d^2 b^4-128 A c^3 d b^4+128 c^3 C d b^4+128 a A d^4 b^3-128 a C d^4 b^3+384 a B c d^3 b^3-384 a A c^2 d^2 b^3+384 a c^2 C d^2 b^3-128 a B c^3 d b^3+i \left (-128 A d^4 b^4+128 C d^4 b^4-384 B c d^3 b^4+384 A c^2 d^2 b^4-384 c^2 C d^2 b^4+128 B c^3 d b^4-128 a B d^4 b^3+384 a A c d^3 b^3-384 a c C d^3 b^3+384 a B c^2 d^2 b^3-128 a A c^3 d b^3+128 a c^3 C d b^3\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 b f}\right )}{4 b}}{6 b}}{8 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}-\frac {\frac {(-a C d-8 b B d+b c C) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 b f}+\frac {-\frac {\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (48 b d^2 (a B+A b-b C)-5 (b c-a d) (-a C d-8 b B d+b c C)\right )}{2 b f}-\frac {3 \left (\frac {\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (64 b^2 d^2 (a A d+a B c-a C d+A b c-b B d-b c C)+(b c-a d) \left (48 b d^2 (a B+A b-b C)-5 (b c-a d) (-a C d-8 b B d+b c C)\right )\right )}{b f}-\frac {\frac {2 \left (5 a^4 C d^4-4 a^3 b d^3 (2 B d+5 c C)+2 a^2 b^2 d^2 \left (8 d^2 (A-C)+20 B c d+15 c^2 C\right )-4 a b^3 d \left (40 c d^2 (A-C)+30 B c^2 d-16 B d^3+5 c^3 C\right )+b^4 \left (-240 c^2 d^2 (A-C)+128 d^4 (A-C)-40 B c^3 d+320 B c d^3+5 c^4 C\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {b} \sqrt {d}}+128 b^3 d \sqrt {a-i b} (c-i d)^{5/2} (B+i (A-C)) \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )-128 b^3 d \sqrt {a+i b} (c+i d)^{5/2} (i A-B-i C) \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{2 b f}\right )}{4 b}}{6 b}}{8 d}\) |
Int[Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x ] + C*Tan[e + f*x]^2),x]
(C*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(7/2))/(4*d*f) - (((b*c*C - 8*b*B*d - a*C*d)*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2))/( 3*b*f) + (-1/2*((48*b*(A*b + a*B - b*C)*d^2 - 5*(b*c - a*d)*(b*c*C - 8*b*B *d - a*C*d))*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(b*f) - (3*(-1/2*(128*Sqrt[a - I*b]*b^3*(B + I*(A - C))*(c - I*d)^(5/2)*d*ArcTanh[ (Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])] - 128*Sqrt[a + I*b]*b^3*(I*A - B - I*C)*(c + I*d)^(5/2)*d*ArcTanh [(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])] + (2*(5*a^4*C*d^4 - 4*a^3*b*d^3*(5*c*C + 2*B*d) + 2*a^2*b^2*d^2* (15*c^2*C + 20*B*c*d + 8*(A - C)*d^2) - 4*a*b^3*d*(5*c^3*C + 30*B*c^2*d + 40*c*(A - C)*d^2 - 16*B*d^3) + b^4*(5*c^4*C - 40*B*c^3*d - 240*c^2*(A - C) *d^2 + 320*B*c*d^3 + 128*(A - C)*d^4))*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[b]*Sqrt[d]))/(b*f) + (( 64*b^2*d^2*(A*b*c + a*B*c - b*c*C + a*A*d - b*B*d - a*C*d) + (b*c - a*d)*( 48*b*(A*b + a*B - b*C)*d^2 - 5*(b*c - a*d)*(b*c*C - 8*b*B*d - a*C*d)))*Sqr t[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(b*f)))/(4*b))/(6*b))/(8*d )
3.2.41.3.1 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[(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_)*((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.
\[\int \sqrt {a +b \tan \left (f x +e \right )}\, \left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}} \left (A +B \tan \left (f x +e \right )+C \tan \left (f x +e \right )^{2}\right )d x\]
Timed out. \[ \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Timed out} \]
integrate((a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C* tan(f*x+e)^2),x, algorithm="fricas")
\[ \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\int \sqrt {a + b \tan {\left (e + f x \right )}} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )\, dx \]
integrate((a+b*tan(f*x+e))**(1/2)*(c+d*tan(f*x+e))**(5/2)*(A+B*tan(f*x+e)+ C*tan(f*x+e)**2),x)
Integral(sqrt(a + b*tan(e + f*x))*(c + d*tan(e + f*x))**(5/2)*(A + B*tan(e + f*x) + C*tan(e + f*x)**2), x)
\[ \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\int { {\left (C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A\right )} \sqrt {b \tan \left (f x + e\right ) + a} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
integrate((a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C* tan(f*x+e)^2),x, algorithm="maxima")
integrate((C*tan(f*x + e)^2 + B*tan(f*x + e) + A)*sqrt(b*tan(f*x + e) + a) *(d*tan(f*x + e) + c)^(5/2), x)
Timed out. \[ \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Timed out} \]
integrate((a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C* tan(f*x+e)^2),x, algorithm="giac")
Timed out. \[ \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\int \sqrt {a+b\,\mathrm {tan}\left (e+f\,x\right )}\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}\,\left (C\,{\mathrm {tan}\left (e+f\,x\right )}^2+B\,\mathrm {tan}\left (e+f\,x\right )+A\right ) \,d x \]
int((a + b*tan(e + f*x))^(1/2)*(c + d*tan(e + f*x))^(5/2)*(A + B*tan(e + f *x) + C*tan(e + f*x)^2),x)