Optimal. Leaf size=563 \[ \frac{\sqrt{\sec (c+d x)} \left (5 a^2 C-2 a b B+2 A b^2-3 b^2 C\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{2 a}{a+b}\right )}{3 b^2 d \left (a^2-b^2\right ) \sqrt{a+b \sec (c+d x)}}-\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (a^2 b^2 (A+9 C)+2 a^3 b B-5 a^4 C-6 a b^3 B+3 A b^4\right )}{3 b^2 d \left (a^2-b^2\right )^2 \sqrt{a+b \sec (c+d x)}}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left (26 a^2 b^2 C+6 a^3 b B-15 a^4 C-14 a b^3 B+8 A b^4-3 b^4 C\right ) \sqrt{a+b \sec (c+d x)}}{3 b^3 d \left (a^2-b^2\right )^2}+\frac{\left (26 a^2 b^2 C+6 a^3 b B-15 a^4 C-14 a b^3 B+8 A b^4-3 b^4 C\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{3 b^3 d \left (a^2-b^2\right )^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{(2 b B-5 a C) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{b^3 d \sqrt{a+b \sec (c+d x)}} \]
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Rubi [A] time = 1.94704, antiderivative size = 563, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 13, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.289, Rules used = {4098, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661} \[ -\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (a^2 b^2 (A+9 C)+2 a^3 b B-5 a^4 C-6 a b^3 B+3 A b^4\right )}{3 b^2 d \left (a^2-b^2\right )^2 \sqrt{a+b \sec (c+d x)}}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left (26 a^2 b^2 C+6 a^3 b B-15 a^4 C-14 a b^3 B+8 A b^4-3 b^4 C\right ) \sqrt{a+b \sec (c+d x)}}{3 b^3 d \left (a^2-b^2\right )^2}+\frac{\sqrt{\sec (c+d x)} \left (5 a^2 C-2 a b B+2 A b^2-3 b^2 C\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{3 b^2 d \left (a^2-b^2\right ) \sqrt{a+b \sec (c+d x)}}+\frac{\left (26 a^2 b^2 C+6 a^3 b B-15 a^4 C-14 a b^3 B+8 A b^4-3 b^4 C\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{3 b^3 d \left (a^2-b^2\right )^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{(2 b B-5 a C) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{b^3 d \sqrt{a+b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 4098
Rule 4102
Rule 4108
Rule 3859
Rule 2807
Rule 2805
Rule 4035
Rule 3856
Rule 2655
Rule 2653
Rule 3858
Rule 2663
Rule 2661
Rubi steps
\begin{align*} \int \frac{\sec ^{\frac{5}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{5/2}} \, dx &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac{2 \int \frac{\sec ^{\frac{3}{2}}(c+d x) \left (\frac{3}{2} \left (A b^2-a (b B-a C)\right )+\frac{3}{2} b (b B-a (A+C)) \sec (c+d x)-\frac{1}{2} \left (2 A b^2-2 a b B+5 a^2 C-3 b^2 C\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{3/2}} \, dx}{3 b \left (a^2-b^2\right )}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}+\frac{2 \left (3 A b^4+2 a^3 b B-6 a b^3 B-5 a^4 C+a^2 b^2 (A+9 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt{a+b \sec (c+d x)}}+\frac{4 \int \frac{\sqrt{\sec (c+d x)} \left (\frac{1}{4} \left (3 A b^4+2 a^3 b B-6 a b^3 B-5 a^4 C+a^2 b^2 (A+9 C)\right )+\frac{1}{4} b \left (a^2 b B+3 b^3 B+2 a^3 C-2 a b^2 (2 A+3 C)\right ) \sec (c+d x)-\frac{1}{4} \left (8 A b^4+6 a^3 b B-14 a b^3 B-15 a^4 C+26 a^2 b^2 C-3 b^4 C\right ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{3 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}+\frac{2 \left (3 A b^4+2 a^3 b B-6 a b^3 B-5 a^4 C+a^2 b^2 (A+9 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt{a+b \sec (c+d x)}}-\frac{\left (8 A b^4+6 a^3 b B-14 a b^3 B-15 a^4 C+26 a^2 b^2 C-3 b^4 C\right ) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d}+\frac{4 \int \frac{\frac{1}{8} a \left (8 A b^4+6 a^3 b B-14 a b^3 B-15 a^4 C+26 a^2 b^2 C-3 b^4 C\right )+\frac{1}{4} b \left (3 A b^4+2 a^3 b B-6 a b^3 B-5 a^4 C+a^2 b^2 (A+9 C)\right ) \sec (c+d x)+\frac{3}{8} \left (a^2-b^2\right )^2 (2 b B-5 a C) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx}{3 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}+\frac{2 \left (3 A b^4+2 a^3 b B-6 a b^3 B-5 a^4 C+a^2 b^2 (A+9 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt{a+b \sec (c+d x)}}-\frac{\left (8 A b^4+6 a^3 b B-14 a b^3 B-15 a^4 C+26 a^2 b^2 C-3 b^4 C\right ) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d}+\frac{4 \int \frac{\frac{1}{8} a \left (8 A b^4+6 a^3 b B-14 a b^3 B-15 a^4 C+26 a^2 b^2 C-3 b^4 C\right )+\frac{1}{4} b \left (3 A b^4+2 a^3 b B-6 a b^3 B-5 a^4 C+a^2 b^2 (A+9 C)\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx}{3 b^3 \left (a^2-b^2\right )^2}+\frac{(2 b B-5 a C) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{2 b^3}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}+\frac{2 \left (3 A b^4+2 a^3 b B-6 a b^3 B-5 a^4 C+a^2 b^2 (A+9 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt{a+b \sec (c+d x)}}-\frac{\left (8 A b^4+6 a^3 b B-14 a b^3 B-15 a^4 C+26 a^2 b^2 C-3 b^4 C\right ) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d}+\frac{\left (2 A b^2-2 a b B+5 a^2 C-3 b^2 C\right ) \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+b \sec (c+d x)}} \, dx}{6 b^2 \left (a^2-b^2\right )}+\frac{\left (8 A b^4+6 a^3 b B-14 a b^3 B-15 a^4 C+26 a^2 b^2 C-3 b^4 C\right ) \int \frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx}{6 b^3 \left (a^2-b^2\right )^2}+\frac{\left ((2 b B-5 a C) \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sec (c+d x)}{\sqrt{b+a \cos (c+d x)}} \, dx}{2 b^3 \sqrt{a+b \sec (c+d x)}}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}+\frac{2 \left (3 A b^4+2 a^3 b B-6 a b^3 B-5 a^4 C+a^2 b^2 (A+9 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt{a+b \sec (c+d x)}}-\frac{\left (8 A b^4+6 a^3 b B-14 a b^3 B-15 a^4 C+26 a^2 b^2 C-3 b^4 C\right ) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d}+\frac{\left (\left (2 A b^2-2 a b B+5 a^2 C-3 b^2 C\right ) \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{b+a \cos (c+d x)}} \, dx}{6 b^2 \left (a^2-b^2\right ) \sqrt{a+b \sec (c+d x)}}+\frac{\left ((2 b B-5 a C) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \sqrt{\sec (c+d x)}\right ) \int \frac{\sec (c+d x)}{\sqrt{\frac{b}{a+b}+\frac{a \cos (c+d x)}{a+b}}} \, dx}{2 b^3 \sqrt{a+b \sec (c+d x)}}+\frac{\left (\left (8 A b^4+6 a^3 b B-14 a b^3 B-15 a^4 C+26 a^2 b^2 C-3 b^4 C\right ) \sqrt{a+b \sec (c+d x)}\right ) \int \sqrt{b+a \cos (c+d x)} \, dx}{6 b^3 \left (a^2-b^2\right )^2 \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)}}\\ &=\frac{(2 b B-5 a C) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right ) \sqrt{\sec (c+d x)}}{b^3 d \sqrt{a+b \sec (c+d x)}}-\frac{2 \left (A b^2-a (b B-a C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}+\frac{2 \left (3 A b^4+2 a^3 b B-6 a b^3 B-5 a^4 C+a^2 b^2 (A+9 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt{a+b \sec (c+d x)}}-\frac{\left (8 A b^4+6 a^3 b B-14 a b^3 B-15 a^4 C+26 a^2 b^2 C-3 b^4 C\right ) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d}+\frac{\left (\left (2 A b^2-2 a b B+5 a^2 C-3 b^2 C\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\frac{b}{a+b}+\frac{a \cos (c+d x)}{a+b}}} \, dx}{6 b^2 \left (a^2-b^2\right ) \sqrt{a+b \sec (c+d x)}}+\frac{\left (\left (8 A b^4+6 a^3 b B-14 a b^3 B-15 a^4 C+26 a^2 b^2 C-3 b^4 C\right ) \sqrt{a+b \sec (c+d x)}\right ) \int \sqrt{\frac{b}{a+b}+\frac{a \cos (c+d x)}{a+b}} \, dx}{6 b^3 \left (a^2-b^2\right )^2 \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \sqrt{\sec (c+d x)}}\\ &=\frac{\left (2 A b^2-2 a b B+5 a^2 C-3 b^2 C\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right ) \sqrt{\sec (c+d x)}}{3 b^2 \left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}+\frac{(2 b B-5 a C) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right ) \sqrt{\sec (c+d x)}}{b^3 d \sqrt{a+b \sec (c+d x)}}+\frac{\left (8 A b^4+6 a^3 b B-14 a b^3 B-15 a^4 C+26 a^2 b^2 C-3 b^4 C\right ) E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right ) \sqrt{a+b \sec (c+d x)}}{3 b^3 \left (a^2-b^2\right )^2 d \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \sqrt{\sec (c+d x)}}-\frac{2 \left (A b^2-a (b B-a C)\right ) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}+\frac{2 \left (3 A b^4+2 a^3 b B-6 a b^3 B-5 a^4 C+a^2 b^2 (A+9 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt{a+b \sec (c+d x)}}-\frac{\left (8 A b^4+6 a^3 b B-14 a b^3 B-15 a^4 C+26 a^2 b^2 C-3 b^4 C\right ) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d}\\ \end{align*}
Mathematica [C] time = 7.34284, size = 938, normalized size = 1.67 \[ \frac{(b+a \cos (c+d x))^3 \sqrt{\sec (c+d x)} \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (-\frac{4 \left (C \sin (c+d x) a^3-b B \sin (c+d x) a^2+A b^2 \sin (c+d x) a\right )}{3 b^2 \left (b^2-a^2\right ) (b+a \cos (c+d x))^2}-\frac{4 \left (-6 C \sin (c+d x) a^5+3 b B \sin (c+d x) a^4+10 b^2 C \sin (c+d x) a^3-7 b^3 B \sin (c+d x) a^2+4 A b^4 \sin (c+d x) a\right )}{3 b^3 \left (b^2-a^2\right )^2 (b+a \cos (c+d x))}+\frac{2 C \tan (c+d x)}{b^3}\right )}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^{5/2}}-\frac{(b+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)} \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (\frac{2 \left (-12 A b^5+24 a B b^4-4 a^2 A b^3-36 a^2 C b^3-8 a^3 B b^2+20 a^4 C b\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{2 a}{a+b}\right )}{\sqrt{b+a \cos (c+d x)}}+\frac{2 \left (45 C a^5-18 b B a^4-86 b^2 C a^3+38 b^3 B a^2-8 A b^4 a+33 b^4 C a-12 b^5 B\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{\sqrt{b+a \cos (c+d x)}}+\frac{2 i \left (15 C a^5-6 b B a^4-26 b^2 C a^3+14 b^3 B a^2-8 A b^4 a+3 b^4 C a\right ) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{\cos (c+d x) a+a}{a-b}} \cos (2 (c+d x)) \left (a \left (2 b \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right ),\frac{b-a}{a+b}\right )+a \Pi \left (1-\frac{a}{b};i \sinh ^{-1}\left (\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right )|\frac{b-a}{a+b}\right )\right )-2 b (a+b) E\left (i \sinh ^{-1}\left (\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right )|\frac{b-a}{a+b}\right )\right ) \sin (c+d x)}{\sqrt{\frac{1}{a-b}} b \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left (-a^2+2 b^2+2 (b+a \cos (c+d x))^2-4 b (b+a \cos (c+d x))\right )}\right )}{6 (a-b)^2 b^3 (a+b)^2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.615, size = 9944, normalized size = 17.7 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [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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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 \frac{{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac{5}{2}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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