Optimal. Leaf size=200 \[ \frac {2 \left (7 a^2+5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{21 d}+\frac {2 \left (7 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {4 a b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {12 a b \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}-\frac {12 a b \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d} \]
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Rubi [A] time = 0.15, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3788, 3768, 3771, 2639, 4046, 2641} \[ \frac {2 \left (7 a^2+5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{21 d}+\frac {2 \left (7 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {4 a b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {12 a b \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}-\frac {12 a b \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 3768
Rule 3771
Rule 3788
Rule 4046
Rubi steps
\begin {align*} \int \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 \, dx &=(2 a b) \int \sec ^{\frac {7}{2}}(c+d x) \, dx+\int \sec ^{\frac {5}{2}}(c+d x) \left (a^2+b^2 \sec ^2(c+d x)\right ) \, dx\\ &=\frac {4 a b \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 b^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {1}{5} (6 a b) \int \sec ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{7} \left (7 a^2+5 b^2\right ) \int \sec ^{\frac {5}{2}}(c+d x) \, dx\\ &=\frac {12 a b \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 \left (7 a^2+5 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {4 a b \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 b^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}-\frac {1}{5} (6 a b) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (7 a^2+5 b^2\right ) \int \sqrt {\sec (c+d x)} \, dx\\ &=\frac {12 a b \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 \left (7 a^2+5 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {4 a b \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 b^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}-\frac {1}{5} \left (6 a b \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} \left (\left (7 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {12 a b \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (7 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {12 a b \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 \left (7 a^2+5 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {4 a b \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 b^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.99, size = 139, normalized size = 0.70 \[ \frac {\sec ^{\frac {7}{2}}(c+d x) \left (20 \left (7 a^2+5 b^2\right ) \cos ^{\frac {7}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+2 \sin (c+d x) \left (5 \left (7 a^2+5 b^2\right ) \cos (2 (c+d x))+35 a^2+273 a b \cos (c+d x)+63 a b \cos (3 (c+d x))+55 b^2\right )-504 a b \cos ^{\frac {7}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{210 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.21, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} \sec \left (d x + c\right )^{4} + 2 \, a b \sec \left (d x + c\right )^{3} + a^{2} \sec \left (d x + c\right )^{2}\right )} \sqrt {\sec \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sec \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 10.77, size = 689, normalized size = 3.44 \[ \text {result too large to display} \]
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 \[ \int {\left (b \sec \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^2\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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