Optimal. Leaf size=309 \[ \frac {2 \left (35 a^2 A-77 a b B-45 A b^2\right ) \sqrt {a+b \tan (c+d x)}}{105 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (105 a^3 B+245 a^2 A b-161 a b^2 B-15 A b^3\right ) \sqrt {a+b \tan (c+d x)}}{105 a d \sqrt {\tan (c+d x)}}+\frac {(-b+i a)^{5/2} (-B+i A) \tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 a (7 a B+10 A b) \sqrt {a+b \tan (c+d x)}}{35 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {(b+i a)^{5/2} (B+i A) \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 a A (a+b \tan (c+d x))^{3/2}}{7 d \tan ^{\frac {7}{2}}(c+d x)} \]
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Rubi [A] time = 1.52, antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {3605, 3645, 3649, 3616, 3615, 93, 203, 206} \[ \frac {2 \left (35 a^2 A-77 a b B-45 A b^2\right ) \sqrt {a+b \tan (c+d x)}}{105 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (245 a^2 A b+105 a^3 B-161 a b^2 B-15 A b^3\right ) \sqrt {a+b \tan (c+d x)}}{105 a d \sqrt {\tan (c+d x)}}+\frac {(-b+i a)^{5/2} (-B+i A) \tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 a (7 a B+10 A b) \sqrt {a+b \tan (c+d x)}}{35 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {(b+i a)^{5/2} (B+i A) \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 a A (a+b \tan (c+d x))^{3/2}}{7 d \tan ^{\frac {7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 93
Rule 203
Rule 206
Rule 3605
Rule 3615
Rule 3616
Rule 3645
Rule 3649
Rubi steps
\begin {align*} \int \frac {(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\tan ^{\frac {9}{2}}(c+d x)} \, dx &=-\frac {2 a A (a+b \tan (c+d x))^{3/2}}{7 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {2}{7} \int \frac {\sqrt {a+b \tan (c+d x)} \left (\frac {1}{2} a (10 A b+7 a B)-\frac {7}{2} \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)-\frac {1}{2} b (4 a A-7 b B) \tan ^2(c+d x)\right )}{\tan ^{\frac {7}{2}}(c+d x)} \, dx\\ &=-\frac {2 a (10 A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 a A (a+b \tan (c+d x))^{3/2}}{7 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {4}{35} \int \frac {-\frac {1}{4} a \left (35 a^2 A-45 A b^2-77 a b B\right )-\frac {35}{4} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)-\frac {1}{4} b \left (60 a A b+28 a^2 B-35 b^2 B\right ) \tan ^2(c+d x)}{\tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx\\ &=-\frac {2 a (10 A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2 A-45 A b^2-77 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 a A (a+b \tan (c+d x))^{3/2}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {8 \int \frac {\frac {1}{8} a \left (245 a^2 A b-15 A b^3+105 a^3 B-161 a b^2 B\right )-\frac {105}{8} a \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \tan (c+d x)-\frac {1}{4} a b \left (35 a^2 A-45 A b^2-77 a b B\right ) \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx}{105 a}\\ &=-\frac {2 a (10 A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2 A-45 A b^2-77 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (245 a^2 A b-15 A b^3+105 a^3 B-161 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{105 a d \sqrt {\tan (c+d x)}}-\frac {2 a A (a+b \tan (c+d x))^{3/2}}{7 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {16 \int \frac {\frac {105}{16} a^2 \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )+\frac {105}{16} a^2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{105 a^2}\\ &=-\frac {2 a (10 A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2 A-45 A b^2-77 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (245 a^2 A b-15 A b^3+105 a^3 B-161 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{105 a d \sqrt {\tan (c+d x)}}-\frac {2 a A (a+b \tan (c+d x))^{3/2}}{7 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {1}{2} \left ((a-i b)^3 (A-i B)\right ) \int \frac {1+i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx+\frac {1}{2} \left ((a+i b)^3 (A+i B)\right ) \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx\\ &=-\frac {2 a (10 A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2 A-45 A b^2-77 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (245 a^2 A b-15 A b^3+105 a^3 B-161 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{105 a d \sqrt {\tan (c+d x)}}-\frac {2 a A (a+b \tan (c+d x))^{3/2}}{7 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {\left ((a-i b)^3 (A-i B)\right ) \operatorname {Subst}\left (\int \frac {1}{(1-i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {\left ((a+i b)^3 (A+i B)\right ) \operatorname {Subst}\left (\int \frac {1}{(1+i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=-\frac {2 a (10 A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2 A-45 A b^2-77 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (245 a^2 A b-15 A b^3+105 a^3 B-161 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{105 a d \sqrt {\tan (c+d x)}}-\frac {2 a A (a+b \tan (c+d x))^{3/2}}{7 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {\left ((a-i b)^3 (A-i B)\right ) \operatorname {Subst}\left (\int \frac {1}{1-(i a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {\left ((a+i b)^3 (A+i B)\right ) \operatorname {Subst}\left (\int \frac {1}{1-(-i a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}\\ &=\frac {(i a-b)^{5/2} (i A-B) \tan ^{-1}\left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {(i a+b)^{5/2} (i A+B) \tanh ^{-1}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 a (10 A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2 A-45 A b^2-77 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (245 a^2 A b-15 A b^3+105 a^3 B-161 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{105 a d \sqrt {\tan (c+d x)}}-\frac {2 a A (a+b \tan (c+d x))^{3/2}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A] time = 6.37, size = 381, normalized size = 1.23 \[ \frac {8 a \left (35 a^2 A-77 a b B-45 A b^2\right ) \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}-6 a \left (28 a^2 B+60 a A b-35 b^2 B\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}-5 a \left (24 a^2 A-49 a b B-28 A b^2\right ) \sqrt {a+b \tan (c+d x)}+8 \left (105 a^3 B+245 a^2 A b-161 a b^2 B-15 A b^3\right ) \tan ^3(c+d x) \sqrt {a+b \tan (c+d x)}+420 \sqrt [4]{-1} a \tan ^{\frac {7}{2}}(c+d x) \left ((-a+i b)^{5/2} (B+i A) \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+(a+i b)^{5/2} (B-i A) \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )\right )-35 a b (a B+4 A b) \sqrt {a+b \tan (c+d x)}-210 a b B (a+b \tan (c+d x))^{3/2}}{420 a d \tan ^{\frac {7}{2}}(c+d x)} \]
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 [B] time = 1.19, size = 2654465, normalized size = 8590.50 \[ \text {output too large to display} \]
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 \[ \int \frac {\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2}}{{\mathrm {tan}\left (c+d\,x\right )}^{9/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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