Optimal. Leaf size=262 \[ \frac {4 \sqrt {2} \sqrt {\pi } b^{5/2} \cos (a) (e (c+d x))^{2/3} C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{5 d (c+d x)^{2/3}}-\frac {4 \sqrt {2} \sqrt {\pi } b^{5/2} \sin (a) (e (c+d x))^{2/3} S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{5 d (c+d x)^{2/3}}-\frac {4 b^2 (e (c+d x))^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{5 d \sqrt [3]{c+d x}}+\frac {3 (c+d x) (e (c+d x))^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{5 d}+\frac {2 b \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{5 d} \]
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Rubi [A] time = 0.23, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3435, 3417, 3415, 3409, 3387, 3388, 3354, 3352, 3351} \[ \frac {4 \sqrt {2} \sqrt {\pi } b^{5/2} \cos (a) (e (c+d x))^{2/3} \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b}}{\sqrt [3]{c+d x}}\right )}{5 d (c+d x)^{2/3}}-\frac {4 \sqrt {2} \sqrt {\pi } b^{5/2} \sin (a) (e (c+d x))^{2/3} S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{5 d (c+d x)^{2/3}}-\frac {4 b^2 (e (c+d x))^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{5 d \sqrt [3]{c+d x}}+\frac {3 (c+d x) (e (c+d x))^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{5 d}+\frac {2 b \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{5 d} \]
Antiderivative was successfully verified.
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Rule 3351
Rule 3352
Rule 3354
Rule 3387
Rule 3388
Rule 3409
Rule 3415
Rule 3417
Rule 3435
Rubi steps
\begin {align*} \int (c e+d e x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx &=\frac {\operatorname {Subst}\left (\int (e x)^{2/3} \sin \left (a+\frac {b}{x^{2/3}}\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {(e (c+d x))^{2/3} \operatorname {Subst}\left (\int x^{2/3} \sin \left (a+\frac {b}{x^{2/3}}\right ) \, dx,x,c+d x\right )}{d (c+d x)^{2/3}}\\ &=\frac {\left (3 (e (c+d x))^{2/3}\right ) \operatorname {Subst}\left (\int x^4 \sin \left (a+\frac {b}{x^2}\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d (c+d x)^{2/3}}\\ &=-\frac {\left (3 (e (c+d x))^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\sin \left (a+b x^2\right )}{x^6} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d (c+d x)^{2/3}}\\ &=\frac {3 (c+d x) (e (c+d x))^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{5 d}-\frac {\left (6 b (e (c+d x))^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\cos \left (a+b x^2\right )}{x^4} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{5 d (c+d x)^{2/3}}\\ &=\frac {2 b \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{5 d}+\frac {3 (c+d x) (e (c+d x))^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{5 d}+\frac {\left (4 b^2 (e (c+d x))^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\sin \left (a+b x^2\right )}{x^2} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{5 d (c+d x)^{2/3}}\\ &=\frac {2 b \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{5 d}-\frac {4 b^2 (e (c+d x))^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{5 d \sqrt [3]{c+d x}}+\frac {3 (c+d x) (e (c+d x))^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{5 d}+\frac {\left (8 b^3 (e (c+d x))^{2/3}\right ) \operatorname {Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{5 d (c+d x)^{2/3}}\\ &=\frac {2 b \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{5 d}-\frac {4 b^2 (e (c+d x))^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{5 d \sqrt [3]{c+d x}}+\frac {3 (c+d x) (e (c+d x))^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{5 d}+\frac {\left (8 b^3 (e (c+d x))^{2/3} \cos (a)\right ) \operatorname {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{5 d (c+d x)^{2/3}}-\frac {\left (8 b^3 (e (c+d x))^{2/3} \sin (a)\right ) \operatorname {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{5 d (c+d x)^{2/3}}\\ &=\frac {2 b \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{5 d}+\frac {4 \sqrt {2} b^{5/2} \sqrt {\pi } (e (c+d x))^{2/3} \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{5 d (c+d x)^{2/3}}-\frac {4 \sqrt {2} b^{5/2} \sqrt {\pi } (e (c+d x))^{2/3} S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{5 d (c+d x)^{2/3}}-\frac {4 b^2 (e (c+d x))^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{5 d \sqrt [3]{c+d x}}+\frac {3 (c+d x) (e (c+d x))^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 228, normalized size = 0.87 \[ \frac {(e (c+d x))^{2/3} \left (4 \sqrt {2 \pi } b^{5/2} \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )-4 \sqrt {2 \pi } b^{5/2} \sin (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )-4 b^2 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )+3 c (c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )+3 d x (c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )+2 b c \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )+2 b d x \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )}{5 d (c+d x)^{2/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d e x + c e\right )}^{\frac {2}{3}} \sin \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {1}{3}} b}{d x + c}\right ), x\right ) \]
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 \[ \int {\left (d e x + c e\right )}^{\frac {2}{3}} \sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \left (d e x +c e \right )^{\frac {2}{3}} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 2.05, size = 749, normalized size = 2.86 \[ \text {result too large to display} \]
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 \[ \int \sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{2/3}}\right )\,{\left (c\,e+d\,e\,x\right )}^{2/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e \left (c + d x\right )\right )^{\frac {2}{3}} \sin {\left (a + \frac {b}{\left (c + d x\right )^{\frac {2}{3}}} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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