Optimal. Leaf size=262 \[ \frac {2 b \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{5 d}+\frac {4 \sqrt {2} b^{5/2} \sqrt {\pi } (e (c+d x))^{2/3} \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{5 d (c+d x)^{2/3}}-\frac {4 \sqrt {2} b^{5/2} \sqrt {\pi } (e (c+d x))^{2/3} S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{5 d (c+d x)^{2/3}}-\frac {4 b^2 (e (c+d x))^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{5 d \sqrt [3]{c+d x}}+\frac {3 (c+d x) (e (c+d x))^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{5 d} \]
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Rubi [A]
time = 0.17, 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 = {3516, 3498,
3496, 3490, 3468, 3469, 3435, 3433, 3432} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 3432
Rule 3433
Rule 3435
Rule 3468
Rule 3469
Rule 3490
Rule 3496
Rule 3498
Rule 3516
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 {\text {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} \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 0.23, size = 228, normalized size = 0.87 \begin {gather*} \frac {(e (c+d x))^{2/3} \left (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 )+4 b^{5/2} \sqrt {2 \pi } \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )-4 b^{5/2} \sqrt {2 \pi } S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)-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 )\right )}{5 d (c+d x)^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, 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] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.83, size = 823, normalized size = 3.14 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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