Optimal. Leaf size=258 \[ -\frac {2 \left (6 a^2-25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{105 b d}-\frac {4 a \left (3 a^2-41 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (6 a^4-31 a^2 b^2+25 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^2 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}-\frac {4 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.39, antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2791, 2753, 2752, 2663, 2661, 2655, 2653} \[ -\frac {2 \left (6 a^2-25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{105 b d}+\frac {2 \left (-31 a^2 b^2+6 a^4+25 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^2 d \sqrt {a+b \cos (c+d x)}}-\frac {4 a \left (3 a^2-41 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}-\frac {4 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2753
Rule 2791
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \, dx &=\frac {2 (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}+\frac {2 \int \left (\frac {5 b}{2}-a \cos (c+d x)\right ) (a+b \cos (c+d x))^{3/2} \, dx}{7 b}\\ &=-\frac {4 a (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 b d}+\frac {2 (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}+\frac {4 \int \sqrt {a+b \cos (c+d x)} \left (\frac {19 a b}{4}-\frac {1}{4} \left (6 a^2-25 b^2\right ) \cos (c+d x)\right ) \, dx}{35 b}\\ &=-\frac {2 \left (6 a^2-25 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b d}-\frac {4 a (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 b d}+\frac {2 (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}+\frac {8 \int \frac {\frac {1}{8} b \left (51 a^2+25 b^2\right )-\frac {1}{4} a \left (3 a^2-41 b^2\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{105 b}\\ &=-\frac {2 \left (6 a^2-25 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b d}-\frac {4 a (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 b d}+\frac {2 (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}+\frac {1}{105} \left (2 a \left (41-\frac {3 a^2}{b^2}\right )\right ) \int \sqrt {a+b \cos (c+d x)} \, dx+\frac {\left (6 a^4-31 a^2 b^2+25 b^4\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{105 b^2}\\ &=-\frac {2 \left (6 a^2-25 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b d}-\frac {4 a (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 b d}+\frac {2 (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}+\frac {\left (2 a \left (41-\frac {3 a^2}{b^2}\right ) \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{105 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (\left (6 a^4-31 a^2 b^2+25 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{105 b^2 \sqrt {a+b \cos (c+d x)}}\\ &=\frac {4 a \left (41-\frac {3 a^2}{b^2}\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (6 a^4-31 a^2 b^2+25 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^2 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (6 a^2-25 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b d}-\frac {4 a (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 b d}+\frac {2 (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.15, size = 214, normalized size = 0.83 \[ \frac {4 \left (6 a^4-31 a^2 b^2+25 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+b \sin (c+d x) \left (12 a^3+b \left (108 a^2+145 b^2\right ) \cos (c+d x)+78 a b^2 \cos (2 (c+d x))+178 a b^2+15 b^3 \cos (3 (c+d x))\right )-8 a \left (3 a^3+3 a^2 b-41 a b^2-41 b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{210 b^2 d \sqrt {a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.33, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \cos \left (d x + c\right )^{3} + a \cos \left (d x + c\right )^{2}\right )} \sqrt {b \cos \left (d x + c\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.70, size = 827, normalized size = 3.21 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\cos \left (c+d\,x\right )}^2\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________