Optimal. Leaf size=383 \[ \frac{\left (3 a^2 C d^2-6 a b d (c C-2 B d)+b^2 \left (8 d^2 (A-C)-4 B c d+3 c^2 C\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{4 \sqrt{b} d^{5/2} f}-\frac{(a-i b)^{3/2} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f \sqrt{c-i d}}+\frac{(a+i b)^{3/2} (i A-B-i C) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f \sqrt{c+i d}}-\frac{(-3 a C d-4 b B d+3 b c C) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 d f} \]
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Rubi [A] time = 4.07667, antiderivative size = 383, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 49, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {3647, 3655, 6725, 63, 217, 206, 93, 208} \[ \frac{\left (3 a^2 C d^2-6 a b d (c C-2 B d)+b^2 \left (8 d^2 (A-C)-4 B c d+3 c^2 C\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{4 \sqrt{b} d^{5/2} f}-\frac{(a-i b)^{3/2} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f \sqrt{c-i d}}+\frac{(a+i b)^{3/2} (i A-B-i C) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f \sqrt{c+i d}}-\frac{(-3 a C d-4 b B d+3 b c C) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 d f} \]
Antiderivative was successfully verified.
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Rule 3647
Rule 3655
Rule 6725
Rule 63
Rule 217
Rule 206
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt{c+d \tan (e+f x)}} \, dx &=\frac{C (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 d f}+\frac{\int \frac{\sqrt{a+b \tan (e+f x)} \left (\frac{1}{2} (-3 b c C+a (4 A-C) d)+2 (A b+a B-b C) d \tan (e+f x)-\frac{1}{2} (3 b c C-4 b B d-3 a C d) \tan ^2(e+f x)\right )}{\sqrt{c+d \tan (e+f x)}} \, dx}{2 d}\\ &=-\frac{(3 b c C-4 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 d f}+\frac{\int \frac{\frac{1}{4} \left (a^2 (8 A-5 C) d^2+b^2 c (3 c C-4 B d)-2 a b d (3 c C+2 B d)\right )+2 \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \tan (e+f x)+\frac{1}{4} \left (8 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-4 b B d-3 a C d)\right ) \tan ^2(e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{2 d^2}\\ &=-\frac{(3 b c C-4 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 d f}+\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{4} \left (a^2 (8 A-5 C) d^2+b^2 c (3 c C-4 B d)-2 a b d (3 c C+2 B d)\right )+2 \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 x+\frac{1}{4} \left (8 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-4 b B d-3 a C d)\right ) x^2}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{2 d^2 f}\\ &=-\frac{(3 b c C-4 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 d f}+\frac{\operatorname{Subst}\left (\int \left (\frac{3 a^2 C d^2-6 a b d (c C-2 B d)+b^2 \left (3 c^2 C-4 B c d+8 (A-C) d^2\right )}{4 \sqrt{a+b x} \sqrt{c+d x}}+\frac{2 \left (-\left (2 a b B-a^2 (A-C)+b^2 (A-C)\right ) d^2+\left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 x\right )}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{2 d^2 f}\\ &=-\frac{(3 b c C-4 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 d f}+\frac{\operatorname{Subst}\left (\int \frac{-\left (2 a b B-a^2 (A-C)+b^2 (A-C)\right ) d^2+\left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 x}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{d^2 f}+\frac{\left (3 a^2 C d^2-6 a b d (c C-2 B d)+b^2 \left (3 c^2 C-4 B c d+8 (A-C) d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{8 d^2 f}\\ &=-\frac{(3 b c C-4 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 d f}+\frac{\operatorname{Subst}\left (\int \left (\frac{-\left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2-i \left (2 a b B-a^2 (A-C)+b^2 (A-C)\right ) d^2}{2 (i-x) \sqrt{a+b x} \sqrt{c+d x}}+\frac{\left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2-i \left (2 a b B-a^2 (A-C)+b^2 (A-C)\right ) d^2}{2 (i+x) \sqrt{a+b x} \sqrt{c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{d^2 f}+\frac{\left (3 a^2 C d^2-6 a b d (c C-2 B d)+b^2 \left (3 c^2 C-4 B c d+8 (A-C) d^2\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 d^2 f}\\ &=-\frac{(3 b c C-4 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 d f}+\frac{\left ((a-i b)^2 (i A+B-i C)\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 f}-\frac{\left (\left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2+i \left (2 a b B-a^2 (A-C)+b^2 (A-C)\right ) d^2\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 d^2 f}+\frac{\left (3 a^2 C d^2-6 a b d (c C-2 B d)+b^2 \left (3 c^2 C-4 B c d+8 (A-C) d^2\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 d^2 f}\\ &=\frac{\left (3 a^2 C d^2-6 a b d (c C-2 B d)+b^2 \left (3 c^2 C-4 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 \sqrt{b} d^{5/2} f}-\frac{(3 b c C-4 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 d f}+\frac{\left ((a-i b)^2 (i A+B-i C)\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 )}{f}-\frac{\left (\left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2+i \left (2 a b B-a^2 (A-C)+b^2 (A-C)\right ) d^2\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 )}{d^2 f}\\ &=-\frac{(a-i b)^{3/2} (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 )}{\sqrt{c-i d} f}+\frac{(a+i b)^{3/2} (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 )}{\sqrt{c+i d} f}+\frac{\left (3 a^2 C d^2-6 a b d (c C-2 B d)+b^2 \left (3 c^2 C-4 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 \sqrt{b} d^{5/2} f}-\frac{(3 b c C-4 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 d f}\\ \end{align*}
Mathematica [A] time = 7.45282, size = 607, normalized size = 1.58 \[ \frac{\frac{\frac{\sqrt{b} \sqrt{c-\frac{a d}{b}} \left (3 a^2 C d^2-6 a b d (c C-2 B d)+b^2 \left (8 d^2 (A-C)-4 B c d+3 c^2 C\right )\right ) \sqrt{\frac{b c+b d \tan (e+f x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c-\frac{a d}{b}}}\right )}{2 \sqrt{d} \sqrt{c+d \tan (e+f x)}}+\frac{2 d^2 \left (b \left (a^2 B+2 a b (A-C)-b^2 B\right )-\sqrt{-b^2} \left (a^2 (-(A-C))+2 a b B+b^2 (A-C)\right )\right ) \tan ^{-1}\left (\frac{\sqrt{\frac{b d}{\sqrt{-b^2}}+c} \sqrt{a+b \tan (e+f x)}}{\sqrt{\sqrt{-b^2}-a} \sqrt{c+d \tan (e+f x)}}\right )}{\sqrt{\sqrt{-b^2}-a} \sqrt{\frac{b d}{\sqrt{-b^2}}+c}}-\frac{2 d^2 \left (b \left (a^2 B+2 a b (A-C)-b^2 B\right )+\sqrt{-b^2} \left (a^2 (-(A-C))+2 a b B+b^2 (A-C)\right )\right ) \tan ^{-1}\left (\frac{\sqrt{-\frac{\sqrt{-b^2} d+b c}{b}} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+\sqrt{-b^2}} \sqrt{c+d \tan (e+f x)}}\right )}{\sqrt{a+\sqrt{-b^2}} \sqrt{-\frac{\sqrt{-b^2} d+b c}{b}}}}{b d f}+\frac{(3 a C d+4 b B d-3 b c C) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{2 d f}}{2 d}+\frac{C (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 d f} \]
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}) \left ( a+b\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{c+d\tan \left ( fx+e \right ) }}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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{\left (a + b \tan{\left (e + f x \right )}\right )^{\frac{3}{2}} \left (A + B \tan{\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\sqrt{c + d \tan{\left (e + f x \right )}}}\, 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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