Optimal. Leaf size=182 \[ -\frac {b^2 \tan (c+d x) \sqrt {b \tan ^4(c+d x)}}{3 d}+\frac {b^2 \tan ^7(c+d x) \sqrt {b \tan ^4(c+d x)}}{9 d}-\frac {b^2 \tan ^5(c+d x) \sqrt {b \tan ^4(c+d x)}}{7 d}+\frac {b^2 \tan ^3(c+d x) \sqrt {b \tan ^4(c+d x)}}{5 d}-b^2 x \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}+\frac {b^2 \cot (c+d x) \sqrt {b \tan ^4(c+d x)}}{d} \]
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Rubi [A] time = 0.06, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3658, 3473, 8} \[ \frac {b^2 \tan ^7(c+d x) \sqrt {b \tan ^4(c+d x)}}{9 d}-\frac {b^2 \tan ^5(c+d x) \sqrt {b \tan ^4(c+d x)}}{7 d}+\frac {b^2 \tan ^3(c+d x) \sqrt {b \tan ^4(c+d x)}}{5 d}-\frac {b^2 \tan (c+d x) \sqrt {b \tan ^4(c+d x)}}{3 d}-b^2 x \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}+\frac {b^2 \cot (c+d x) \sqrt {b \tan ^4(c+d x)}}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rule 3658
Rubi steps
\begin {align*} \int \left (b \tan ^4(c+d x)\right )^{5/2} \, dx &=\left (b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int \tan ^{10}(c+d x) \, dx\\ &=\frac {b^2 \tan ^7(c+d x) \sqrt {b \tan ^4(c+d x)}}{9 d}-\left (b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int \tan ^8(c+d x) \, dx\\ &=-\frac {b^2 \tan ^5(c+d x) \sqrt {b \tan ^4(c+d x)}}{7 d}+\frac {b^2 \tan ^7(c+d x) \sqrt {b \tan ^4(c+d x)}}{9 d}+\left (b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int \tan ^6(c+d x) \, dx\\ &=\frac {b^2 \tan ^3(c+d x) \sqrt {b \tan ^4(c+d x)}}{5 d}-\frac {b^2 \tan ^5(c+d x) \sqrt {b \tan ^4(c+d x)}}{7 d}+\frac {b^2 \tan ^7(c+d x) \sqrt {b \tan ^4(c+d x)}}{9 d}-\left (b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int \tan ^4(c+d x) \, dx\\ &=-\frac {b^2 \tan (c+d x) \sqrt {b \tan ^4(c+d x)}}{3 d}+\frac {b^2 \tan ^3(c+d x) \sqrt {b \tan ^4(c+d x)}}{5 d}-\frac {b^2 \tan ^5(c+d x) \sqrt {b \tan ^4(c+d x)}}{7 d}+\frac {b^2 \tan ^7(c+d x) \sqrt {b \tan ^4(c+d x)}}{9 d}+\left (b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int \tan ^2(c+d x) \, dx\\ &=\frac {b^2 \cot (c+d x) \sqrt {b \tan ^4(c+d x)}}{d}-\frac {b^2 \tan (c+d x) \sqrt {b \tan ^4(c+d x)}}{3 d}+\frac {b^2 \tan ^3(c+d x) \sqrt {b \tan ^4(c+d x)}}{5 d}-\frac {b^2 \tan ^5(c+d x) \sqrt {b \tan ^4(c+d x)}}{7 d}+\frac {b^2 \tan ^7(c+d x) \sqrt {b \tan ^4(c+d x)}}{9 d}-\left (b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\right ) \int 1 \, dx\\ &=\frac {b^2 \cot (c+d x) \sqrt {b \tan ^4(c+d x)}}{d}-b^2 x \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}-\frac {b^2 \tan (c+d x) \sqrt {b \tan ^4(c+d x)}}{3 d}+\frac {b^2 \tan ^3(c+d x) \sqrt {b \tan ^4(c+d x)}}{5 d}-\frac {b^2 \tan ^5(c+d x) \sqrt {b \tan ^4(c+d x)}}{7 d}+\frac {b^2 \tan ^7(c+d x) \sqrt {b \tan ^4(c+d x)}}{9 d}\\ \end {align*}
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Mathematica [A] time = 0.76, size = 86, normalized size = 0.47 \[ \frac {\cot (c+d x) \left (b \tan ^4(c+d x)\right )^{5/2} \left (315 \cot ^8(c+d x)-105 \cot ^6(c+d x)+63 \cot ^4(c+d x)-45 \cot ^2(c+d x)-315 \tan ^{-1}(\tan (c+d x)) \cot ^9(c+d x)+35\right )}{315 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.11, size = 96, normalized size = 0.53 \[ \frac {{\left (35 \, b^{2} \tan \left (d x + c\right )^{9} - 45 \, b^{2} \tan \left (d x + c\right )^{7} + 63 \, b^{2} \tan \left (d x + c\right )^{5} - 105 \, b^{2} \tan \left (d x + c\right )^{3} - 315 \, b^{2} d x + 315 \, b^{2} \tan \left (d x + c\right )\right )} \sqrt {b \tan \left (d x + c\right )^{4}}}{315 \, d \tan \left (d x + c\right )^{2}} \]
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 [A] time = 0.12, size = 84, normalized size = 0.46 \[ -\frac {\left (b \left (\tan ^{4}\left (d x +c \right )\right )\right )^{\frac {5}{2}} \left (-35 \left (\tan ^{9}\left (d x +c \right )\right )+45 \left (\tan ^{7}\left (d x +c \right )\right )-63 \left (\tan ^{5}\left (d x +c \right )\right )+105 \left (\tan ^{3}\left (d x +c \right )\right )+315 \arctan \left (\tan \left (d x +c \right )\right )-315 \tan \left (d x +c \right )\right )}{315 d \tan \left (d x +c \right )^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.84, size = 79, normalized size = 0.43 \[ \frac {35 \, b^{\frac {5}{2}} \tan \left (d x + c\right )^{9} - 45 \, b^{\frac {5}{2}} \tan \left (d x + c\right )^{7} + 63 \, b^{\frac {5}{2}} \tan \left (d x + c\right )^{5} - 105 \, b^{\frac {5}{2}} \tan \left (d x + c\right )^{3} - 315 \, {\left (d x + c\right )} b^{\frac {5}{2}} + 315 \, b^{\frac {5}{2}} \tan \left (d x + c\right )}{315 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^4\right )}^{5/2} \,d x \]
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 \[ \int \left (b \tan ^{4}{\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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