Optimal. Leaf size=505 \[ \frac {\left (5 a^3 C d^3-15 a^2 b d^2 (c C-2 B d)+5 a b^2 d \left (8 d^2 (A-C)-4 B c d+3 c^2 C\right )-\left (b^3 \left (8 c d^2 (A-C)-6 B c^2 d+16 B d^3+5 c^3 C\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{8 \sqrt {b} d^{7/2} f}+\frac {\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (8 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-6 b B d+5 b c C)\right )}{8 d^3 f}-\frac {(a-i b)^{5/2} (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {c-i d}}-\frac {(a+i b)^{5/2} (B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {c+i d}}-\frac {(-5 a C d-6 b B d+5 b c C) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{12 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 d f} \]
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Rubi [A] time = 5.95, antiderivative size = 505, normalized size of antiderivative = 1.00, number of steps used = 15, 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} \[ \frac {\left (-15 a^2 b d^2 (c C-2 B d)+5 a^3 C d^3+5 a b^2 d \left (8 d^2 (A-C)-4 B c d+3 c^2 C\right )+b^3 \left (-\left (8 c d^2 (A-C)-6 B c^2 d+16 B d^3+5 c^3 C\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{8 \sqrt {b} d^{7/2} f}+\frac {\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (8 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-6 b B d+5 b c C)\right )}{8 d^3 f}-\frac {(a-i b)^{5/2} (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {c-i d}}-\frac {(a+i b)^{5/2} (B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {c+i d}}-\frac {(-5 a C d-6 b B d+5 b c C) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{12 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 d f} \]
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 {(a+b \tan (e+f x))^{5/2} \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))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 d f}+\frac {\int \frac {(a+b \tan (e+f x))^{3/2} \left (\frac {1}{2} (-5 b c C+a (6 A-C) d)+3 (A b+a B-b C) d \tan (e+f x)-\frac {1}{2} (5 b c C-6 b B d-5 a C d) \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx}{3 d}\\ &=-\frac {(5 b c C-6 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{12 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 d f}+\frac {\int \frac {\sqrt {a+b \tan (e+f x)} \left (\frac {1}{4} (-4 a d (5 b c C-a (6 A-C) d)+(3 b c+a d) (5 b c C-6 b B d-5 a C d))+6 \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \tan (e+f x)+\frac {3}{4} \left (8 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-6 b B d-5 a C d)\right ) \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx}{6 d^2}\\ &=\frac {\left (8 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-6 b B d-5 a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 d^3 f}-\frac {(5 b c C-6 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{12 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 d f}+\frac {\int \frac {\frac {3}{8} \left (a^3 (16 A-11 C) d^3-3 a^2 b d^2 (5 c C+6 B d)+a b^2 d \left (15 c^2 C-20 B c d-8 (A-C) d^2\right )-b^3 c \left (5 c^2 C-6 B c d+8 (A-C) d^2\right )\right )+6 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^3 \tan (e+f x)+\frac {3}{8} \left (16 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3-(b c-a d) \left (8 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-6 b B d-5 a C d)\right )\right ) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{6 d^3}\\ &=\frac {\left (8 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-6 b B d-5 a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 d^3 f}-\frac {(5 b c C-6 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{12 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 d f}+\frac {\operatorname {Subst}\left (\int \frac {\frac {3}{8} \left (a^3 (16 A-11 C) d^3-3 a^2 b d^2 (5 c C+6 B d)+a b^2 d \left (15 c^2 C-20 B c d-8 (A-C) d^2\right )-b^3 c \left (5 c^2 C-6 B c d+8 (A-C) d^2\right )\right )+6 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^3 x+\frac {3}{8} \left (16 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3-(b c-a d) \left (8 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-6 b B d-5 a C d)\right )\right ) x^2}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{6 d^3 f}\\ &=\frac {\left (8 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-6 b B d-5 a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 d^3 f}-\frac {(5 b c C-6 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{12 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 d f}+\frac {\operatorname {Subst}\left (\int \left (\frac {3 \left (5 a^3 C d^3-15 a^2 b d^2 (c C-2 B d)+5 a b^2 d \left (3 c^2 C-4 B c d+8 (A-C) d^2\right )-b^3 \left (5 c^3 C-6 B c^2 d+8 c (A-C) d^2+16 B d^3\right )\right )}{8 \sqrt {a+b x} \sqrt {c+d x}}+\frac {6 \left (-\left (3 a^2 b B-b^3 B-a^3 (A-C)+3 a b^2 (A-C)\right ) d^3+\left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^3 x\right )}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{6 d^3 f}\\ &=\frac {\left (8 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-6 b B d-5 a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 d^3 f}-\frac {(5 b c C-6 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{12 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 d f}+\frac {\operatorname {Subst}\left (\int \frac {-\left (3 a^2 b B-b^3 B-a^3 (A-C)+3 a b^2 (A-C)\right ) d^3+\left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^3 x}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{d^3 f}+\frac {\left (5 a^3 C d^3-15 a^2 b d^2 (c C-2 B d)+5 a b^2 d \left (3 c^2 C-4 B c d+8 (A-C) d^2\right )-b^3 \left (5 c^3 C-6 B c^2 d+8 c (A-C) d^2+16 B d^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{16 d^3 f}\\ &=\frac {\left (8 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-6 b B d-5 a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 d^3 f}-\frac {(5 b c C-6 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{12 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 d f}+\frac {\operatorname {Subst}\left (\int \left (\frac {-i \left (3 a^2 b B-b^3 B-a^3 (A-C)+3 a b^2 (A-C)\right ) d^3-\left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^3}{2 (i-x) \sqrt {a+b x} \sqrt {c+d x}}+\frac {-i \left (3 a^2 b B-b^3 B-a^3 (A-C)+3 a b^2 (A-C)\right ) d^3+\left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^3}{2 (i+x) \sqrt {a+b x} \sqrt {c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{d^3 f}+\frac {\left (5 a^3 C d^3-15 a^2 b d^2 (c C-2 B d)+5 a b^2 d \left (3 c^2 C-4 B c d+8 (A-C) d^2\right )-b^3 \left (5 c^3 C-6 B c^2 d+8 c (A-C) d^2+16 B d^3\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 )}{8 b d^3 f}\\ &=\frac {\left (8 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-6 b B d-5 a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 d^3 f}-\frac {(5 b c C-6 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{12 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 d f}+\frac {\left ((a-i b)^3 (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 (i \left (3 a^2 b B-b^3 B-a^3 (A-C)+3 a b^2 (A-C)\right ) d^3+\left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) 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 d^3 f}+\frac {\left (5 a^3 C d^3-15 a^2 b d^2 (c C-2 B d)+5 a b^2 d \left (3 c^2 C-4 B c d+8 (A-C) d^2\right )-b^3 \left (5 c^3 C-6 B c^2 d+8 c (A-C) d^2+16 B d^3\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 )}{8 b d^3 f}\\ &=\frac {\left (5 a^3 C d^3-15 a^2 b d^2 (c C-2 B d)+5 a b^2 d \left (3 c^2 C-4 B c d+8 (A-C) d^2\right )-b^3 \left (5 c^3 C-6 B c^2 d+8 c (A-C) d^2+16 B d^3\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 )}{8 \sqrt {b} d^{7/2} f}+\frac {\left (8 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-6 b B d-5 a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 d^3 f}-\frac {(5 b c C-6 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{12 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 d f}+\frac {\left ((a-i b)^3 (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 (i \left (3 a^2 b B-b^3 B-a^3 (A-C)+3 a b^2 (A-C)\right ) d^3+\left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) 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 )}{d^3 f}\\ &=-\frac {(a-i b)^{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 )}{\sqrt {c-i d} f}+\frac {(a+i b)^{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 )}{\sqrt {c+i d} f}+\frac {\left (5 a^3 C d^3-15 a^2 b d^2 (c C-2 B d)+5 a b^2 d \left (3 c^2 C-4 B c d+8 (A-C) d^2\right )-b^3 \left (5 c^3 C-6 B c^2 d+8 c (A-C) d^2+16 B d^3\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 )}{8 \sqrt {b} d^{7/2} f}+\frac {\left (8 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-6 b B d-5 a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 d^3 f}-\frac {(5 b c C-6 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{12 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 d f}\\ \end {align*}
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Mathematica [A] time = 8.50, size = 785, normalized size = 1.55 \[ \frac {\frac {\frac {\frac {3 \sqrt {b} \sqrt {c-\frac {a d}{b}} \left (5 a^3 C d^3-15 a^2 b d^2 (c C-2 B d)+5 a b^2 d \left (8 d^2 (A-C)-4 B c d+3 c^2 C\right )-\left (b^3 \left (8 c d^2 (A-C)-6 B c^2 d+16 B d^3+5 c^3 C\right )\right )\right ) \sqrt {\frac {b c+b d \tan (e+f x)}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c-\frac {a d}{b}}}\right )}{4 \sqrt {d} \sqrt {c+d \tan (e+f x)}}-\frac {6 d^3 \left (\sqrt {-b^2} \left (-\left (a^3 (A-C)\right )+3 a^2 b B+3 a b^2 (A-C)-b^3 B\right )-b \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {\frac {\sqrt {-b^2} d}{b}-c} \sqrt {a+b \tan (e+f x)}}{\sqrt {\sqrt {-b^2}-a} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {\sqrt {-b^2}-a} \sqrt {\frac {\sqrt {-b^2} d}{b}-c}}-\frac {6 d^3 \left (\sqrt {-b^2} \left (-\left (a^3 (A-C)\right )+3 a^2 b B+3 a b^2 (A-C)-b^3 B\right )+b \left (a^3 B+3 a^2 b (A-C)-3 a b^2 B-b^3 (A-C)\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {\frac {\sqrt {-b^2} d}{b}+c} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+\sqrt {-b^2}} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+\sqrt {-b^2}} \sqrt {\frac {\sqrt {-b^2} d}{b}+c}}}{b d f}+\frac {3 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (8 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-6 b B d+5 b c C)\right )}{4 d f}}{2 d}+\frac {(5 a C d+6 b B d-5 b c C) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{4 d f}}{3 d}+\frac {C (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}{3 d f} \]
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 \[ \int \frac {\left (a +b \tan \left (f x +e \right )\right )^{\frac {5}{2}} \left (A +B \tan \left (f x +e \right )+C \left (\tan ^{2}\left (f x +e \right )\right )\right )}{\sqrt {c +d \tan \left (f x +e \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}\,\left (C\,{\mathrm {tan}\left (e+f\,x\right )}^2+B\,\mathrm {tan}\left (e+f\,x\right )+A\right )}{\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}} \,d x \]
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 \[ \int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\sqrt {c + d \tan {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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