Integrand size = 25, antiderivative size = 280 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\frac {i (i a-b)^{3/2} \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {a \left (a^2+24 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{8 b^{3/2} d}-\frac {i (i a+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {\left (a^2+8 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{8 b d}-\frac {a \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{12 b d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 b d} \]
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Time = 2.63 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3647, 3728, 3736, 6857, 65, 223, 212, 95, 211, 214} \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=-\frac {a \left (a^2+24 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{8 b^{3/2} d}-\frac {\left (a^2+8 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{8 b d}+\frac {i (-b+i a)^{3/2} \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {i (b+i a)^{3/2} \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 b d}-\frac {a \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{12 b d} \]
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Rule 65
Rule 95
Rule 211
Rule 212
Rule 214
Rule 223
Rule 3647
Rule 3728
Rule 3736
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 b d}+\frac {\int \frac {(a+b \tan (c+d x))^{3/2} \left (-\frac {a}{2}-3 b \tan (c+d x)-\frac {1}{2} a \tan ^2(c+d x)\right )}{\sqrt {\tan (c+d x)}} \, dx}{3 b} \\ & = -\frac {a \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{12 b d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 b d}+\frac {\int \frac {\sqrt {a+b \tan (c+d x)} \left (-\frac {3 a^2}{4}-6 a b \tan (c+d x)-\frac {3}{4} \left (a^2+8 b^2\right ) \tan ^2(c+d x)\right )}{\sqrt {\tan (c+d x)}} \, dx}{6 b} \\ & = -\frac {\left (a^2+8 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{8 b d}-\frac {a \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{12 b d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 b d}+\frac {\int \frac {-\frac {3}{8} a \left (a^2-8 b^2\right )-6 b \left (a^2-b^2\right ) \tan (c+d x)-\frac {3}{8} a \left (a^2+24 b^2\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{6 b} \\ & = -\frac {\left (a^2+8 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{8 b d}-\frac {a \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{12 b d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 b d}+\frac {\text {Subst}\left (\int \frac {-\frac {3}{8} a \left (a^2-8 b^2\right )-6 b \left (a^2-b^2\right ) x-\frac {3}{8} a \left (a^2+24 b^2\right ) x^2}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{6 b d} \\ & = -\frac {\left (a^2+8 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{8 b d}-\frac {a \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{12 b d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 b d}+\frac {\text {Subst}\left (\int \left (-\frac {3 a \left (a^2+24 b^2\right )}{8 \sqrt {x} \sqrt {a+b x}}+\frac {6 \left (2 a b^2-b \left (a^2-b^2\right ) x\right )}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{6 b d} \\ & = -\frac {\left (a^2+8 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{8 b d}-\frac {a \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{12 b d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 b d}+\frac {\text {Subst}\left (\int \frac {2 a b^2-b \left (a^2-b^2\right ) x}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{b d}-\frac {\left (a \left (a^2+24 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{16 b d} \\ & = -\frac {\left (a^2+8 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{8 b d}-\frac {a \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{12 b d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 b d}+\frac {\text {Subst}\left (\int \left (\frac {2 i a b^2+b \left (a^2-b^2\right )}{2 (i-x) \sqrt {x} \sqrt {a+b x}}+\frac {2 i a b^2-b \left (a^2-b^2\right )}{2 \sqrt {x} (i+x) \sqrt {a+b x}}\right ) \, dx,x,\tan (c+d x)\right )}{b d}-\frac {\left (a \left (a^2+24 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {\tan (c+d x)}\right )}{8 b d} \\ & = -\frac {\left (a^2+8 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{8 b d}-\frac {a \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{12 b d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 b d}-\frac {(a-i b)^2 \text {Subst}\left (\int \frac {1}{\sqrt {x} (i+x) \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {(a+i b)^2 \text {Subst}\left (\int \frac {1}{(i-x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}-\frac {\left (a \left (a^2+24 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{8 b d} \\ & = -\frac {a \left (a^2+24 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{8 b^{3/2} d}-\frac {\left (a^2+8 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{8 b d}-\frac {a \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{12 b d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 b d}-\frac {(a-i b)^2 \text {Subst}\left (\int \frac {1}{i-(-a+i b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {(a+i b)^2 \text {Subst}\left (\int \frac {1}{i-(a+i b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d} \\ & = \frac {i (i a-b)^{3/2} \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {a \left (a^2+24 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{8 b^{3/2} d}-\frac {i (i a+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {\left (a^2+8 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{8 b d}-\frac {a \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{12 b d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 b d} \\ \end{align*}
Time = 4.02 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.12 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\frac {-3 a^{3/2} \left (a^2+24 b^2\right ) \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right ) \sqrt {1+\frac {b \tan (c+d x)}{a}}+\sqrt {b} \left (24 \sqrt [4]{-1} (-a+i b)^{3/2} b \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {a+b \tan (c+d x)}-24 \sqrt [4]{-1} (a+i b)^{3/2} b \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {a+b \tan (c+d x)}+\sqrt {\tan (c+d x)} \left (3 \left (a^3-8 a b^2\right )+b \left (17 a^2-24 b^2\right ) \tan (c+d x)+22 a b^2 \tan ^2(c+d x)+8 b^3 \tan ^3(c+d x)\right )\right )}{24 b^{3/2} d \sqrt {a+b \tan (c+d x)}} \]
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result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 0.62 (sec) , antiderivative size = 1347308, normalized size of antiderivative = 4811.81
\[\text {output too large to display}\]
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Leaf count of result is larger than twice the leaf count of optimal. 3676 vs. \(2 (224) = 448\).
Time = 1.22 (sec) , antiderivative size = 7354, normalized size of antiderivative = 26.26 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\text {Timed out} \]
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\[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \tan \left (d x + c\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\int {\mathrm {tan}\left (c+d\,x\right )}^{5/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2} \,d x \]
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