Optimal. Leaf size=384 \[ \frac {\left (3 a^2 C d^2-2 a b d (2 B d+3 c C)+b^2 \left (8 d^2 (A-C)+12 B c d+3 c^2 C\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{4 b^{5/2} \sqrt {d} f}-\frac {(c-i d)^{3/2} (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a-i b}}+\frac {(c+i d)^{3/2} (i A-B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a+i b}}+\frac {(-3 a C d+4 b B d+3 b c C) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b^2 f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f} \]
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Rubi [A] time = 4.31, antiderivative size = 384, normalized size of antiderivative = 1.00, 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} \[ \frac {\left (3 a^2 C d^2-2 a b d (2 B d+3 c C)+b^2 \left (8 d^2 (A-C)+12 B c d+3 c^2 C\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{4 b^{5/2} \sqrt {d} f}-\frac {(c-i d)^{3/2} (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a-i b}}+\frac {(c+i d)^{3/2} (i A-B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a+i b}}+\frac {(-3 a C d+4 b B d+3 b c C) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b^2 f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f} \]
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 206
Rule 208
Rule 217
Rule 3647
Rule 3655
Rule 6725
Rubi steps
\begin {align*} \int \frac {(c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {a+b \tan (e+f x)}} \, dx &=\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}+\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (\frac {1}{2} (4 A b c-C (b c+3 a d))+2 b (B c+(A-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 {a+b \tan (e+f x)}} \, dx}{2 b}\\ &=\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 b^2 f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}+\frac {\int \frac {\frac {1}{4} \left (8 A b^2 c^2+3 a^2 C d^2-2 a b d (3 c C+2 B d)-b^2 c (5 c C+4 B d)\right )+2 b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) \tan (e+f x)+\frac {1}{4} \left (8 b^2 d (B c+(A-C) d)+(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 b^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 b^2 f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}+\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{4} \left (8 A b^2 c^2+3 a^2 C d^2-2 a b d (3 c C+2 B d)-b^2 c (5 c C+4 B d)\right )+2 b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) x+\frac {1}{4} \left (8 b^2 d (B c+(A-C) d)+(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 b^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 b^2 f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}+\frac {\operatorname {Subst}\left (\int \left (\frac {3 a^2 C d^2-2 a b d (3 c C+2 B d)+b^2 \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )}{4 \sqrt {a+b x} \sqrt {c+d x}}+\frac {2 \left (-b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) x\right )}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{2 b^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 b^2 f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}+\frac {\operatorname {Subst}\left (\int \frac {-b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{b^2 f}+\frac {\left (3 a^2 C d^2-2 a b d (3 c C+2 B d)+b^2 \left (3 c^2 C+12 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 b^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 b^2 f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}+\frac {\operatorname {Subst}\left (\int \left (\frac {-i b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )}{2 (i-x) \sqrt {a+b x} \sqrt {c+d x}}+\frac {-i b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )}{2 (i+x) \sqrt {a+b x} \sqrt {c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{b^2 f}+\frac {\left (3 a^2 C d^2-2 a b d (3 c C+2 B d)+b^2 \left (3 c^2 C+12 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^3 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 b^2 f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}+\frac {\left ((i A+B-i C) (c-i 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 f}+\frac {\left (3 a^2 C d^2-2 a b d (3 c C+2 B d)+b^2 \left (3 c^2 C+12 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^3 f}-\frac {\left (i b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\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 b^2 f}\\ &=\frac {\left (3 a^2 C d^2-2 a b d (3 c C+2 B d)+b^2 \left (3 c^2 C+12 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^{5/2} \sqrt {d} 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 b^2 f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}+\frac {\left ((i A+B-i C) (c-i 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 )}{f}-\frac {\left (i b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\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 )}{b^2 f}\\ &=-\frac {(i A+B-i C) (c-i d)^{3/2} \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 )}{\sqrt {a-i b} f}+\frac {(i A-B-i C) (c+i d)^{3/2} \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 )}{\sqrt {a+i b} f}+\frac {\left (3 a^2 C d^2-2 a b d (3 c C+2 B d)+b^2 \left (3 c^2 C+12 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^{5/2} \sqrt {d} 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 b^2 f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}\\ \end {align*}
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Mathematica [A] time = 7.69, size = 613, normalized size = 1.60 \[ \frac {\frac {\frac {\sqrt {b} \sqrt {c-\frac {a d}{b}} \left (3 a^2 C d^2-2 a b d (2 B d+3 c C)+b^2 \left (8 d^2 (A-C)+12 B c d+3 c^2 C\right )\right ) \sqrt {\frac {b c+b d \tan (e+f x)}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c-\frac {a d}{b}}}\right )}{2 \sqrt {d} \sqrt {c+d \tan (e+f x)}}-\frac {2 b^2 \left (\sqrt {-b^2} \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-b \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {\frac {\sqrt {-b^2} d}{b}-c} \sqrt {a+b \tan (e+f x)}}{\sqrt {\sqrt {-b^2}-a} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {\sqrt {-b^2}-a} \sqrt {\frac {\sqrt {-b^2} d}{b}-c}}-\frac {2 b^2 \left (\sqrt {-b^2} \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )+b \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {\frac {\sqrt {-b^2} d}{b}+c} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+\sqrt {-b^2}} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+\sqrt {-b^2}} \sqrt {\frac {\sqrt {-b^2} d}{b}+c}}}{b^2 f}+\frac {(-3 a C d+4 b B d+3 b c C) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{2 b f}}{2 b}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}} \left (A +B \tan \left (f x +e \right )+C \left (\tan ^{2}\left (f x +e \right )\right )\right )}{\sqrt {a +b \tan \left (f x +e \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{\sqrt {b \tan \left (f x + e\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\sqrt {a + b \tan {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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