Optimal. Leaf size=169 \[ \frac{2 b B \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sqrt{b \sec (c+d x)}}{3 d}-\frac{6 b^2 C E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 B \sin (c+d x) (b \sec (c+d x))^{3/2}}{3 d}+\frac{2 C \sin (c+d x) (b \sec (c+d x))^{5/2}}{5 b d}+\frac{6 b C \sin (c+d x) \sqrt{b \sec (c+d x)}}{5 d} \]
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Rubi [A] time = 0.143518, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.219, Rules used = {4047, 3768, 3771, 2641, 12, 16, 2639} \[ -\frac{6 b^2 C E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 B \sin (c+d x) (b \sec (c+d x))^{3/2}}{3 d}+\frac{2 b B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{3 d}+\frac{2 C \sin (c+d x) (b \sec (c+d x))^{5/2}}{5 b d}+\frac{6 b C \sin (c+d x) \sqrt{b \sec (c+d x)}}{5 d} \]
Antiderivative was successfully verified.
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Rule 4047
Rule 3768
Rule 3771
Rule 2641
Rule 12
Rule 16
Rule 2639
Rubi steps
\begin{align*} \int (b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{B \int (b \sec (c+d x))^{5/2} \, dx}{b}+\int C \sec ^2(c+d x) (b \sec (c+d x))^{3/2} \, dx\\ &=\frac{2 B (b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac{1}{3} (b B) \int \sqrt{b \sec (c+d x)} \, dx+C \int \sec ^2(c+d x) (b \sec (c+d x))^{3/2} \, dx\\ &=\frac{2 B (b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac{C \int (b \sec (c+d x))^{7/2} \, dx}{b^2}+\frac{1}{3} \left (b B \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 b B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{3 d}+\frac{2 B (b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac{2 C (b \sec (c+d x))^{5/2} \sin (c+d x)}{5 b d}+\frac{1}{5} (3 C) \int (b \sec (c+d x))^{3/2} \, dx\\ &=\frac{2 b B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{3 d}+\frac{6 b C \sqrt{b \sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 B (b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac{2 C (b \sec (c+d x))^{5/2} \sin (c+d x)}{5 b d}-\frac{1}{5} \left (3 b^2 C\right ) \int \frac{1}{\sqrt{b \sec (c+d x)}} \, dx\\ &=\frac{2 b B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{3 d}+\frac{6 b C \sqrt{b \sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 B (b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac{2 C (b \sec (c+d x))^{5/2} \sin (c+d x)}{5 b d}-\frac{\left (3 b^2 C\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}\\ &=-\frac{6 b^2 C E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 b B \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{3 d}+\frac{6 b C \sqrt{b \sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 B (b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac{2 C (b \sec (c+d x))^{5/2} \sin (c+d x)}{5 b d}\\ \end{align*}
Mathematica [A] time = 0.448616, size = 102, normalized size = 0.6 \[ \frac{(b \sec (c+d x))^{5/2} \left (20 B \cos ^{\frac{5}{2}}(c+d x) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+10 B \sin (2 (c+d x))+21 C \sin (c+d x)+9 C \sin (3 (c+d x))-36 C \cos ^{\frac{5}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{30 b d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.289, size = 526, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} \left (b \sec \left (d x + c\right )\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C b \sec \left (d x + c\right )^{3} + B b \sec \left (d x + c\right )^{2}\right )} \sqrt{b \sec \left (d x + c\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} \left (b \sec \left (d x + c\right )\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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