Integrand size = 35, antiderivative size = 244 \[ \int \frac {\sqrt {\tan (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=-\frac {(i A-B) \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(i a-b)^{5/2} d}+\frac {(i A+B) \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(i a+b)^{5/2} d}-\frac {2 (A b-a B) \sqrt {\tan (c+d x)}}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {2 \left (5 a^2 A b-A b^3-2 a^3 B+4 a b^2 B\right ) \sqrt {\tan (c+d x)}}{3 a \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}} \]
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Time = 1.15 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3689, 3730, 3697, 3696, 95, 209, 212} \[ \int \frac {\sqrt {\tan (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=-\frac {2 (A b-a B) \sqrt {\tan (c+d x)}}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {2 \left (-2 a^3 B+5 a^2 A b+4 a b^2 B-A b^3\right ) \sqrt {\tan (c+d x)}}{3 a d \left (a^2+b^2\right )^2 \sqrt {a+b \tan (c+d x)}}-\frac {(-B+i A) \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d (-b+i a)^{5/2}}+\frac {(B+i A) \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d (b+i a)^{5/2}} \]
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Rule 95
Rule 209
Rule 212
Rule 3689
Rule 3696
Rule 3697
Rule 3730
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (A b-a B) \sqrt {\tan (c+d x)}}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {2 \int \frac {-\frac {1}{2} b (A b-a B)-\frac {3}{2} b (a A+b B) \tan (c+d x)+b (A b-a B) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx}{3 b \left (a^2+b^2\right )} \\ & = -\frac {2 (A b-a B) \sqrt {\tan (c+d x)}}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {2 \left (5 a^2 A b-A b^3-2 a^3 B+4 a b^2 B\right ) \sqrt {\tan (c+d x)}}{3 a \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}-\frac {4 \int \frac {-\frac {3}{4} a b \left (2 a A b-a^2 B+b^2 B\right )-\frac {3}{4} a b \left (a^2 A-A b^2+2 a b B\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{3 a b \left (a^2+b^2\right )^2} \\ & = -\frac {2 (A b-a B) \sqrt {\tan (c+d x)}}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {2 \left (5 a^2 A b-A b^3-2 a^3 B+4 a b^2 B\right ) \sqrt {\tan (c+d x)}}{3 a \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {(i A-B) \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{2 (a+i b)^2}-\frac {(i A+B) \int \frac {1+i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{2 (a-i b)^2} \\ & = -\frac {2 (A b-a B) \sqrt {\tan (c+d x)}}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {2 \left (5 a^2 A b-A b^3-2 a^3 B+4 a b^2 B\right ) \sqrt {\tan (c+d x)}}{3 a \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {(i A-B) \text {Subst}\left (\int \frac {1}{(1+i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 (a+i b)^2 d}-\frac {(i A+B) \text {Subst}\left (\int \frac {1}{(1-i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 (a-i b)^2 d} \\ & = -\frac {2 (A b-a B) \sqrt {\tan (c+d x)}}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {2 \left (5 a^2 A b-A b^3-2 a^3 B+4 a b^2 B\right ) \sqrt {\tan (c+d x)}}{3 a \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {(i A-B) \text {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 )}{(a+i b)^2 d}-\frac {(i A+B) \text {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 )}{(a-i b)^2 d} \\ & = \frac {(i A-B) \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {i a-b} (a+i b)^2 d}+\frac {(i A+B) \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(i a+b)^{5/2} d}-\frac {2 (A b-a B) \sqrt {\tan (c+d x)}}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {2 \left (5 a^2 A b-A b^3-2 a^3 B+4 a b^2 B\right ) \sqrt {\tan (c+d x)}}{3 a \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}} \\ \end{align*}
Time = 3.74 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt {\tan (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\frac {\frac {2 b (A b-a B) \tan ^{\frac {3}{2}}(c+d x)}{(a+b \tan (c+d x))^{3/2}}+\frac {6 b \left (2 a A b-a^2 B+b^2 B\right ) \tan ^{\frac {3}{2}}(c+d x)}{\left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {3 \left (-\frac {\sqrt [4]{-1} a (a+i b)^2 (A-i B) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {-a+i b}}+\frac {\sqrt [4]{-1} a (a-i b)^2 (A+i B) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {a+i b}}+2 \left (-2 a A b+a^2 B-b^2 B\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}\right )}{a^2+b^2}}{3 a \left (a^2+b^2\right ) d} \]
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result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 2.54 (sec) , antiderivative size = 2978176, normalized size of antiderivative = 12205.64
\[\text {output too large to display}\]
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Leaf count of result is larger than twice the leaf count of optimal. 27593 vs. \(2 (200) = 400\).
Time = 14.52 (sec) , antiderivative size = 27593, normalized size of antiderivative = 113.09 \[ \int \frac {\sqrt {\tan (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {\sqrt {\tan (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \sqrt {\tan {\left (c + d x \right )}}}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {\sqrt {\tan (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \sqrt {\tan \left (d x + c\right )}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {\tan (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\sqrt {\tan (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx=\int \frac {\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )}{{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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