Optimal. Leaf size=332 \[ \frac {(i a-b)^{5/2} \text {ArcTan}\left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {\left (5 a^4+240 a^2 b^2-128 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{64 b^{3/2} d}-\frac {(i a+b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {a \left (5 a^2+112 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{64 b d}-\frac {\left (5 a^2+48 b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{96 b d}-\frac {a \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{24 b d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d} \]
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Rubi [A]
time = 1.93, antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3647, 3728,
3736, 6857, 65, 223, 212, 95, 211, 214} \begin {gather*} -\frac {\left (5 a^2+48 b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{96 b d}-\frac {a \left (5 a^2+112 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{64 b d}-\frac {\left (5 a^4+240 a^2 b^2-128 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{64 b^{3/2} d}+\frac {(-b+i a)^{5/2} \text {ArcTan}\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))^{7/2}}{4 b d}-\frac {a \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{24 b d}-\frac {(b+i a)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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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*} \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} \, dx &=\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}+\frac {\int \frac {(a+b \tan (c+d x))^{5/2} \left (-\frac {a}{2}-4 b \tan (c+d x)-\frac {1}{2} a \tan ^2(c+d x)\right )}{\sqrt {\tan (c+d x)}} \, dx}{4 b}\\ &=-\frac {a \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{24 b d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}+\frac {\int \frac {(a+b \tan (c+d x))^{3/2} \left (-\frac {5 a^2}{4}-12 a b \tan (c+d x)-\frac {1}{4} \left (5 a^2+48 b^2\right ) \tan ^2(c+d x)\right )}{\sqrt {\tan (c+d x)}} \, dx}{12 b}\\ &=-\frac {\left (5 a^2+48 b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{96 b d}-\frac {a \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{24 b d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}+\frac {\int \frac {\sqrt {a+b \tan (c+d x)} \left (-\frac {3}{8} a \left (5 a^2-16 b^2\right )-24 b \left (a^2-b^2\right ) \tan (c+d x)-\frac {3}{8} a \left (5 a^2+112 b^2\right ) \tan ^2(c+d x)\right )}{\sqrt {\tan (c+d x)}} \, dx}{24 b}\\ &=-\frac {a \left (5 a^2+112 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{64 b d}-\frac {\left (5 a^2+48 b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{96 b d}-\frac {a \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{24 b d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}+\frac {\int \frac {-\frac {3}{16} a^2 \left (5 a^2-144 b^2\right )-24 a b \left (a^2-3 b^2\right ) \tan (c+d x)-\frac {3}{16} \left (5 a^4+240 a^2 b^2-128 b^4\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{24 b}\\ &=-\frac {a \left (5 a^2+112 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{64 b d}-\frac {\left (5 a^2+48 b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{96 b d}-\frac {a \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{24 b d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}+\frac {\text {Subst}\left (\int \frac {-\frac {3}{16} a^2 \left (5 a^2-144 b^2\right )-24 a b \left (a^2-3 b^2\right ) x-\frac {3}{16} \left (5 a^4+240 a^2 b^2-128 b^4\right ) x^2}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{24 b d}\\ &=-\frac {a \left (5 a^2+112 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{64 b d}-\frac {\left (5 a^2+48 b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{96 b d}-\frac {a \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{24 b d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}+\frac {\text {Subst}\left (\int \left (-\frac {3 \left (5 a^4+240 a^2 b^2-128 b^4\right )}{16 \sqrt {x} \sqrt {a+b x}}+\frac {24 \left (3 a^2 b^2-b^4-a b \left (a^2-3 b^2\right ) x\right )}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{24 b d}\\ &=-\frac {a \left (5 a^2+112 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{64 b d}-\frac {\left (5 a^2+48 b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{96 b d}-\frac {a \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{24 b d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}+\frac {\text {Subst}\left (\int \frac {3 a^2 b^2-b^4-a b \left (a^2-3 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 (5 a^4+240 a^2 b^2-128 b^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{128 b d}\\ &=-\frac {a \left (5 a^2+112 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{64 b d}-\frac {\left (5 a^2+48 b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{96 b d}-\frac {a \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{24 b d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}+\frac {\text {Subst}\left (\int \left (\frac {a b \left (a^2-3 b^2\right )+i \left (3 a^2 b^2-b^4\right )}{2 (i-x) \sqrt {x} \sqrt {a+b x}}+\frac {-a b \left (a^2-3 b^2\right )+i \left (3 a^2 b^2-b^4\right )}{2 \sqrt {x} (i+x) \sqrt {a+b x}}\right ) \, dx,x,\tan (c+d x)\right )}{b d}-\frac {\left (5 a^4+240 a^2 b^2-128 b^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {\tan (c+d x)}\right )}{64 b d}\\ &=-\frac {a \left (5 a^2+112 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{64 b d}-\frac {\left (5 a^2+48 b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{96 b d}-\frac {a \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{24 b d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}-\frac {(a-i b)^3 \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)^3 \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 (5 a^4+240 a^2 b^2-128 b^4\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 )}{64 b d}\\ &=-\frac {\left (5 a^4+240 a^2 b^2-128 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{64 b^{3/2} d}-\frac {a \left (5 a^2+112 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{64 b d}-\frac {\left (5 a^2+48 b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{96 b d}-\frac {a \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{24 b d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}-\frac {(a-i b)^3 \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)^3 \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 a-b)^{5/2} \tan ^{-1}\left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {\left (5 a^4+240 a^2 b^2-128 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{64 b^{3/2} d}-\frac {(i a+b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {a \left (5 a^2+112 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{64 b d}-\frac {\left (5 a^2+48 b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{96 b d}-\frac {a \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{24 b d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}\\ \end {align*}
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Mathematica [A]
time = 3.42, size = 349, normalized size = 1.05 \begin {gather*} -\frac {192 \sqrt [4]{-1} (-a+i b)^{5/2} b \text {ArcTan}\left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+192 \sqrt [4]{-1} (a+i b)^{5/2} b \text {ArcTan}\left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+3 a \left (5 a^2+112 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}+2 \left (5 a^2+48 b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}+8 a \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}-48 \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}+\frac {3 \sqrt {a} \left (5 a^4+240 a^2 b^2-128 b^4\right ) \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right ) \sqrt {1+\frac {b \tan (c+d x)}{a}}}{\sqrt {b} \sqrt {a+b \tan (c+d x)}}}{192 b d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] result has leaf size over 500,000. Avoiding possible recursion issues.
time = 0.50, size = 1347722, normalized size = 4059.40 \[\text {output too large to display}\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int {\mathrm {tan}\left (c+d\,x\right )}^{5/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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