Optimal. Leaf size=286 \[ -\frac {15 a^3 (e \cos (c+d x))^{3/2}}{8 d e \sqrt {a \sin (c+d x)+a}}-\frac {3 a^2 \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{4 d e}+\frac {15 a^2 \sqrt {e} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \tan ^{-1}\left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{8 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac {15 a^2 \sqrt {e} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \sinh ^{-1}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right )}{8 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}{3 d e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.44, antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2678, 2684, 2775, 203, 2833, 63, 215} \[ -\frac {15 a^3 (e \cos (c+d x))^{3/2}}{8 d e \sqrt {a \sin (c+d x)+a}}-\frac {3 a^2 \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{4 d e}+\frac {15 a^2 \sqrt {e} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \tan ^{-1}\left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{8 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac {15 a^2 \sqrt {e} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \sinh ^{-1}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right )}{8 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}{3 d e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 203
Rule 215
Rule 2678
Rule 2684
Rule 2775
Rule 2833
Rubi steps
\begin {align*} \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2} \, dx &=-\frac {a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}{3 d e}+\frac {1}{2} (3 a) \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2} \, dx\\ &=-\frac {3 a^2 (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}{4 d e}-\frac {a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}{3 d e}+\frac {1}{8} \left (15 a^2\right ) \int \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {15 a^3 (e \cos (c+d x))^{3/2}}{8 d e \sqrt {a+a \sin (c+d x)}}-\frac {3 a^2 (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}{4 d e}-\frac {a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}{3 d e}+\frac {1}{16} \left (15 a^3\right ) \int \frac {\sqrt {e \cos (c+d x)}}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=-\frac {15 a^3 (e \cos (c+d x))^{3/2}}{8 d e \sqrt {a+a \sin (c+d x)}}-\frac {3 a^2 (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}{4 d e}-\frac {a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}{3 d e}+\frac {\left (15 a^3 e \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \int \frac {\sqrt {1+\cos (c+d x)}}{\sqrt {e \cos (c+d x)}} \, dx}{16 (a+a \cos (c+d x)+a \sin (c+d x))}-\frac {\left (15 a^3 e \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \int \frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}} \, dx}{16 (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac {15 a^3 (e \cos (c+d x))^{3/2}}{8 d e \sqrt {a+a \sin (c+d x)}}-\frac {3 a^2 (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}{4 d e}-\frac {a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}{3 d e}+\frac {\left (15 a^3 e \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {e x} \sqrt {1+x}} \, dx,x,\cos (c+d x)\right )}{16 d (a+a \cos (c+d x)+a \sin (c+d x))}-\frac {\left (15 a^3 e \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1+e x^2} \, dx,x,-\frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right )}{8 d (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac {15 a^3 (e \cos (c+d x))^{3/2}}{8 d e \sqrt {a+a \sin (c+d x)}}-\frac {3 a^2 (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}{4 d e}-\frac {a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}{3 d e}+\frac {15 a^3 \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{8 d (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {\left (15 a^3 \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{e}}} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{8 d (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac {15 a^3 (e \cos (c+d x))^{3/2}}{8 d e \sqrt {a+a \sin (c+d x)}}-\frac {3 a^2 (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}{4 d e}-\frac {a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}{3 d e}+\frac {15 a^3 \sqrt {e} \sinh ^{-1}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{8 d (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {15 a^3 \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{8 d (a+a \cos (c+d x)+a \sin (c+d x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.12, size = 78, normalized size = 0.27 \[ -\frac {16 \sqrt [4]{2} a (a (\sin (c+d x)+1))^{3/2} (e \cos (c+d x))^{3/2} \, _2F_1\left (-\frac {9}{4},\frac {3}{4};\frac {7}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{3 d e (\sin (c+d x)+1)^{9/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [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]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.34, size = 318, normalized size = 1.11 \[ -\frac {\left (16 \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )-45 \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right ) \sin \left (d x +c \right )-45 \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \sin \left (d x +c \right )+16 \left (\cos ^{4}\left (d x +c \right )\right )-68 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+52 \left (\cos ^{3}\left (d x +c \right )\right )-90 \cos \left (d x +c \right ) \sin \left (d x +c \right )-158 \left (\cos ^{2}\left (d x +c \right )\right )+90 \cos \left (d x +c \right )\right ) \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {5}{2}} \sqrt {e \cos \left (d x +c \right )}}{48 d \left (\left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-\left (\cos ^{3}\left (d x +c \right )\right )+2 \cos \left (d x +c \right ) \sin \left (d x +c \right )+3 \left (\cos ^{2}\left (d x +c \right )\right )-4 \sin \left (d x +c \right )+2 \cos \left (d x +c \right )-4\right ) \cos \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {e\,\cos \left (c+d\,x\right )}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \]
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]
________________________________________________________________________________________