Optimal. Leaf size=370 \[ -\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{c+d \tan (e+f x)}}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}-\frac{2 \sqrt{c+d \tan (e+f x)} \left (-a^2 b^2 (5 A d+3 B c-7 C d)+2 a^3 b B d+a^4 C d+2 a b^3 (3 A c-2 B d-3 c C)+b^4 (A d+3 B c)\right )}{3 b f \left (a^2+b^2\right )^2 (b c-a d) \sqrt{a+b \tan (e+f x)}}-\frac{\sqrt{c-i d} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a-i b)^{5/2}}-\frac{\sqrt{c+i d} (B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a+i b)^{5/2}} \]
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Rubi [A] time = 2.05186, antiderivative size = 370, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 49, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.122, Rules used = {3645, 3649, 3616, 3615, 93, 208} \[ -\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{c+d \tan (e+f x)}}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}-\frac{2 \sqrt{c+d \tan (e+f x)} \left (-a^2 b^2 (5 A d+3 B c-7 C d)+2 a^3 b B d+a^4 C d+2 a b^3 (3 A c-2 B d-3 c C)+b^4 (A d+3 B c)\right )}{3 b f \left (a^2+b^2\right )^2 (b c-a d) \sqrt{a+b \tan (e+f x)}}-\frac{\sqrt{c-i d} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a-i b)^{5/2}}-\frac{\sqrt{c+i d} (B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a+i b)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3645
Rule 3649
Rule 3616
Rule 3615
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{5/2}} \, dx &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac{2 \int \frac{\frac{1}{2} ((b B-a C) (3 b c-a d)+A b (3 a c+b d))-\frac{3}{2} b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)-\frac{1}{2} \left (2 A b^2-2 a b B-a^2 C-3 b^2 C\right ) d \tan ^2(e+f x)}{(a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}} \, dx}{3 b \left (a^2+b^2\right )}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (2 a^3 b B d+a^4 C d+b^4 (3 B c+A d)+2 a b^3 (3 A c-3 c C-2 B d)-a^2 b^2 (3 B c+5 A d-7 C d)\right ) \sqrt{c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right )^2 (b c-a d) f \sqrt{a+b \tan (e+f x)}}-\frac{4 \int \frac{-\frac{3}{4} b (b c-a d) \left (a^2 (A c-c C-B d)-b^2 (A c-c C-B d)+2 a b (B c+(A-C) d)\right )+\frac{3}{4} b (b c-a d) \left (2 a b (A c-c C-B d)-a^2 (B c+(A-C) d)+b^2 (B c+(A-C) d)\right ) \tan (e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{3 b \left (a^2+b^2\right )^2 (b c-a d)}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (2 a^3 b B d+a^4 C d+b^4 (3 B c+A d)+2 a b^3 (3 A c-3 c C-2 B d)-a^2 b^2 (3 B c+5 A d-7 C d)\right ) \sqrt{c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right )^2 (b c-a d) f \sqrt{a+b \tan (e+f x)}}+\frac{((A-i B-C) (c-i d)) \int \frac{1+i \tan (e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{2 (a-i b)^2}+\frac{((A+i B-C) (c+i d)) \int \frac{1-i \tan (e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{2 (a+i b)^2}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (2 a^3 b B d+a^4 C d+b^4 (3 B c+A d)+2 a b^3 (3 A c-3 c C-2 B d)-a^2 b^2 (3 B c+5 A d-7 C d)\right ) \sqrt{c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right )^2 (b c-a d) f \sqrt{a+b \tan (e+f x)}}+\frac{((A-i B-C) (c-i d)) \operatorname{Subst}\left (\int \frac{1}{(1-i x) \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (a-i b)^2 f}+\frac{((A+i B-C) (c+i d)) \operatorname{Subst}\left (\int \frac{1}{(1+i x) \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (a+i b)^2 f}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (2 a^3 b B d+a^4 C d+b^4 (3 B c+A d)+2 a b^3 (3 A c-3 c C-2 B d)-a^2 b^2 (3 B c+5 A d-7 C d)\right ) \sqrt{c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right )^2 (b c-a d) f \sqrt{a+b \tan (e+f x)}}+\frac{((A-i B-C) (c-i d)) \operatorname{Subst}\left (\int \frac{1}{i a+b-(i c+d) x^2} \, dx,x,\frac{\sqrt{a+b \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}}\right )}{(a-i b)^2 f}+\frac{((A+i B-C) (c+i d)) \operatorname{Subst}\left (\int \frac{1}{-i a+b-(-i c+d) x^2} \, dx,x,\frac{\sqrt{a+b \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}}\right )}{(a+i b)^2 f}\\ &=-\frac{(i A+B-i C) \sqrt{c-i d} \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)^{5/2} f}-\frac{(B-i (A-C)) \sqrt{c+i d} \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)^{5/2} f}-\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (2 a^3 b B d+a^4 C d+b^4 (3 B c+A d)+2 a b^3 (3 A c-3 c C-2 B d)-a^2 b^2 (3 B c+5 A d-7 C d)\right ) \sqrt{c+d \tan (e+f x)}}{3 b \left (a^2+b^2\right )^2 (b c-a d) f \sqrt{a+b \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 6.95865, size = 603, normalized size = 1.63 \[ -\frac{C \sqrt{c+d \tan (e+f x)}}{b f (a+b \tan (e+f x))^{3/2}}-\frac{-\frac{2 \sqrt{c+d \tan (e+f x)} \left (\frac{1}{2} b^2 (-a C d-2 A b c+3 b c C)-a \left (b^2 (-(d (A-C)+B c))-\frac{1}{2} a (-a C d-2 b B d+b c C)\right )\right )}{3 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (-\frac{2 \sqrt{c+d \tan (e+f x)} \left (\frac{1}{2} b^2 (b c-a d) \left (a^2 C d+a b (3 A c-B d-3 c C)+b^2 (A d+3 B c)\right )-a \left (\frac{1}{2} a d (b c-a d) \left (a^2 (-C)-2 a b B+2 A b^2-3 b^2 C\right )-\frac{3}{2} b^2 (b c-a d) (-a A d-a B c+a C d+A b c-b B d-b c C)\right )\right )}{f \left (a^2+b^2\right ) (b c-a d) \sqrt{a+b \tan (e+f x)}}-\frac{3 b (b c-a d) \left (\frac{(a-i b)^2 \sqrt{-c-i d} (i A-B-i C) \tan ^{-1}\left (\frac{\sqrt{-c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{\sqrt{a+i b}}-\frac{(a+i b)^2 \sqrt{c-i d} (B+i (A-C)) \tan ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{\sqrt{-a+i b}}\right )}{2 f \left (a^2+b^2\right )}\right )}{3 \left (a^2+b^2\right ) (b c-a d)}}{b} \]
Antiderivative was successfully verified.
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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})\sqrt{c+d\tan \left ( fx+e \right ) } \left ( a+b\tan \left ( fx+e \right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c + d \tan{\left (e + f x \right )}} \left (A + B \tan{\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\left (a + b \tan{\left (e + f x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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