Optimal. Leaf size=535 \[ \frac{\sqrt{d} \left (15 a^2 C d^2-6 a b d (2 B d+5 c C)+b^2 \left (8 d^2 (A-C)+20 B c d+15 c^2 C\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{4 b^{7/2} f}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) \sqrt{a+b \tan (e+f x)}}+\frac{d \left (5 a^2 C-4 a b B+4 A b^2+b^2 C\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b^2 f \left (a^2+b^2\right )}-\frac{d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left (-3 a^2 b (4 B d+5 c C)+15 a^3 C d-8 A b^2 (b c-a d)+a b^2 (8 B c+7 C d)-b^3 (4 B d+7 c C)\right )}{4 b^3 f \left (a^2+b^2\right )}-\frac{(c-i d)^{5/2} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a-i b)^{3/2}}-\frac{(c+i d)^{5/2} (B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a+i b)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 8.31384, antiderivative size = 535, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 49, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.184, Rules used = {3645, 3647, 3655, 6725, 63, 217, 206, 93, 208} \[ \frac{\sqrt{d} \left (15 a^2 C d^2-6 a b d (2 B d+5 c C)+b^2 \left (8 d^2 (A-C)+20 B c d+15 c^2 C\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{4 b^{7/2} f}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) \sqrt{a+b \tan (e+f x)}}+\frac{d \left (5 a^2 C-4 a b B+4 A b^2+b^2 C\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b^2 f \left (a^2+b^2\right )}-\frac{d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left (-3 a^2 b (4 B d+5 c C)+15 a^3 C d-8 A b^2 (b c-a d)+a b^2 (8 B c+7 C d)-b^3 (4 B d+7 c C)\right )}{4 b^3 f \left (a^2+b^2\right )}-\frac{(c-i d)^{5/2} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a-i b)^{3/2}}-\frac{(c+i d)^{5/2} (B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a+i b)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3645
Rule 3647
Rule 3655
Rule 6725
Rule 63
Rule 217
Rule 206
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{3/2}} \, dx &=-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}+\frac{2 \int \frac{(c+d \tan (e+f x))^{3/2} \left (\frac{1}{2} ((b B-a C) (b c-5 a d)+A b (a c+5 b d))-\frac{1}{2} b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+\frac{1}{2} \left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) d \tan ^2(e+f x)\right )}{\sqrt{a+b \tan (e+f x)}} \, dx}{b \left (a^2+b^2\right )}\\ &=\frac{\left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) d \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b^2 \left (a^2+b^2\right ) f}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}+\frac{\int \frac{\sqrt{c+d \tan (e+f x)} \left (\frac{1}{4} \left (-\left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) d (b c+3 a d)+4 b c ((b B-a C) (b c-5 a d)+A b (a c+5 b d))\right )+b^2 \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^2 C+2 B c d-C d^2\right )\right ) \tan (e+f x)+\frac{1}{4} d \left (3 \left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) (b c-a d)-4 b^2 ((A-C) (b c-a d)-B (a c+b d))\right ) \tan ^2(e+f x)\right )}{\sqrt{a+b \tan (e+f x)}} \, dx}{b^2 \left (a^2+b^2\right )}\\ &=-\frac{d \left (15 a^3 C d-8 A b^2 (b c-a d)-3 a^2 b (5 c C+4 B d)-b^3 (7 c C+4 B d)+a b^2 (8 B c+7 C d)\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 b^3 \left (a^2+b^2\right ) f}+\frac{\left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) d \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b^2 \left (a^2+b^2\right ) f}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}+\frac{\int \frac{\frac{1}{8} \left (15 a^4 C d^3-6 a^3 b d^2 (5 c C+2 B d)+b^4 c \left (8 B c^2+24 A c d-9 c C d-4 B d^2\right )+a^2 b^2 d \left (15 c^2 C+20 B c d+(8 A+7 C) d^2\right )-2 a b^3 \left (4 c^3 C+12 B c^2 d+3 c C d^2+2 B d^3-4 A \left (c^3-3 c d^2\right )\right )\right )+b^3 \left (a A d \left (3 c^2-d^2\right )-A b \left (c^3-3 c d^2\right )+b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )-a \left (C d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)+\frac{1}{8} \left (a^2+b^2\right ) d \left (15 a^2 C d^2-6 a b d (5 c C+2 B d)+b^2 \left (15 c^2 C+20 B c d+8 (A-C) d^2\right )\right ) \tan ^2(e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{b^3 \left (a^2+b^2\right )}\\ &=-\frac{d \left (15 a^3 C d-8 A b^2 (b c-a d)-3 a^2 b (5 c C+4 B d)-b^3 (7 c C+4 B d)+a b^2 (8 B c+7 C d)\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 b^3 \left (a^2+b^2\right ) f}+\frac{\left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) d \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b^2 \left (a^2+b^2\right ) f}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}+\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{8} \left (15 a^4 C d^3-6 a^3 b d^2 (5 c C+2 B d)+b^4 c \left (8 B c^2+24 A c d-9 c C d-4 B d^2\right )+a^2 b^2 d \left (15 c^2 C+20 B c d+(8 A+7 C) d^2\right )-2 a b^3 \left (4 c^3 C+12 B c^2 d+3 c C d^2+2 B d^3-4 A \left (c^3-3 c d^2\right )\right )\right )+b^3 \left (a A d \left (3 c^2-d^2\right )-A b \left (c^3-3 c d^2\right )+b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )-a \left (C d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )\right ) x+\frac{1}{8} \left (a^2+b^2\right ) d \left (15 a^2 C d^2-6 a b d (5 c C+2 B d)+b^2 \left (15 c^2 C+20 B c d+8 (A-C) d^2\right )\right ) x^2}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{b^3 \left (a^2+b^2\right ) f}\\ &=-\frac{d \left (15 a^3 C d-8 A b^2 (b c-a d)-3 a^2 b (5 c C+4 B d)-b^3 (7 c C+4 B d)+a b^2 (8 B c+7 C d)\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 b^3 \left (a^2+b^2\right ) f}+\frac{\left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) d \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b^2 \left (a^2+b^2\right ) f}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}+\frac{\operatorname{Subst}\left (\int \left (\frac{\left (a^2+b^2\right ) d \left (15 a^2 C d^2-6 a b d (5 c C+2 B d)+b^2 \left (15 c^2 C+20 B c d+8 (A-C) d^2\right )\right )}{8 \sqrt{a+b x} \sqrt{c+d x}}+\frac{-b^3 \left (a \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )+b^3 \left (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 )+A \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right )\right ) x}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{b^3 \left (a^2+b^2\right ) f}\\ &=-\frac{d \left (15 a^3 C d-8 A b^2 (b c-a d)-3 a^2 b (5 c C+4 B d)-b^3 (7 c C+4 B d)+a b^2 (8 B c+7 C d)\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 b^3 \left (a^2+b^2\right ) f}+\frac{\left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) d \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b^2 \left (a^2+b^2\right ) f}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}+\frac{\operatorname{Subst}\left (\int \frac{-b^3 \left (a \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )+b^3 \left (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 )+A \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right )\right ) x}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{b^3 \left (a^2+b^2\right ) f}+\frac{\left (d \left (15 a^2 C d^2-6 a b d (5 c C+2 B d)+b^2 \left (15 c^2 C+20 B c d+8 (A-C) d^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{8 b^3 f}\\ &=-\frac{d \left (15 a^3 C d-8 A b^2 (b c-a d)-3 a^2 b (5 c C+4 B d)-b^3 (7 c C+4 B d)+a b^2 (8 B c+7 C d)\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 b^3 \left (a^2+b^2\right ) f}+\frac{\left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) d \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b^2 \left (a^2+b^2\right ) f}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}+\frac{\operatorname{Subst}\left (\int \left (\frac{-b^3 \left (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 )+A \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right )\right )-i b^3 \left (a \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )}{2 (i-x) \sqrt{a+b x} \sqrt{c+d x}}+\frac{b^3 \left (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 )+A \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right )\right )-i b^3 \left (a \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )}{2 (i+x) \sqrt{a+b x} \sqrt{c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{b^3 \left (a^2+b^2\right ) f}+\frac{\left (d \left (15 a^2 C d^2-6 a b d (5 c C+2 B d)+b^2 \left (15 c^2 C+20 B c d+8 (A-C) d^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b \tan (e+f x)}\right )}{4 b^4 f}\\ &=-\frac{d \left (15 a^3 C d-8 A b^2 (b c-a d)-3 a^2 b (5 c C+4 B d)-b^3 (7 c C+4 B d)+a b^2 (8 B c+7 C d)\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 b^3 \left (a^2+b^2\right ) f}+\frac{\left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) d \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b^2 \left (a^2+b^2\right ) f}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}+\frac{\left ((i A+B-i C) (c-i d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{(i+x) \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (a-i b) f}+\frac{\left ((i a+b) (A+i B-C) (c+i d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{(i-x) \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 \left (a^2+b^2\right ) f}+\frac{\left (d \left (15 a^2 C d^2-6 a b d (5 c C+2 B d)+b^2 \left (15 c^2 C+20 B c d+8 (A-C) d^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}}\right )}{4 b^4 f}\\ &=\frac{\sqrt{d} \left (15 a^2 C d^2-6 a b d (5 c C+2 B d)+b^2 \left (15 c^2 C+20 B c d+8 (A-C) d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{4 b^{7/2} f}-\frac{d \left (15 a^3 C d-8 A b^2 (b c-a d)-3 a^2 b (5 c C+4 B d)-b^3 (7 c C+4 B d)+a b^2 (8 B c+7 C d)\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 b^3 \left (a^2+b^2\right ) f}+\frac{\left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) d \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b^2 \left (a^2+b^2\right ) f}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}+\frac{\left ((i A+B-i C) (c-i d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{-a+i b-(-c+i d) x^2} \, dx,x,\frac{\sqrt{a+b \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}}\right )}{(a-i b) f}+\frac{\left ((i a+b) (A+i B-C) (c+i d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{a+i b-(c+i d) x^2} \, dx,x,\frac{\sqrt{a+b \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}}\right )}{\left (a^2+b^2\right ) f}\\ &=-\frac{(i A+B-i C) (c-i d)^{5/2} \tanh ^{-1}\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 )}{(a-i b)^{3/2} f}-\frac{(B-i (A-C)) (c+i d)^{5/2} \tanh ^{-1}\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 )}{(a+i b)^{3/2} f}+\frac{\sqrt{d} \left (15 a^2 C d^2-6 a b d (5 c C+2 B d)+b^2 \left (15 c^2 C+20 B c d+8 (A-C) d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{4 b^{7/2} f}-\frac{d \left (15 a^3 C d-8 A b^2 (b c-a d)-3 a^2 b (5 c C+4 B d)-b^3 (7 c C+4 B d)+a b^2 (8 B c+7 C d)\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 b^3 \left (a^2+b^2\right ) f}+\frac{\left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) d \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b^2 \left (a^2+b^2\right ) f}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}\\ \end{align*}
Mathematica [C] time = 44.0091, size = 1654245, normalized size = 3092.05 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{(A+B\tan \left ( fx+e \right ) +C \left ( \tan \left ( fx+e \right ) \right ) ^{2}) \left ( c+d\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}} \left ( a+b\tan \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]