Optimal. Leaf size=299 \[ -\frac {8 \sqrt {2 \pi } b^{7/2} e \sin (a) \sqrt [3]{e (c+d x)} C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{35 d \sqrt [3]{c+d x}}-\frac {8 \sqrt {2 \pi } b^{7/2} e \cos (a) \sqrt [3]{e (c+d x)} S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{35 d \sqrt [3]{c+d x}}-\frac {8 b^3 e \sqrt [3]{e (c+d x)} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{35 d}-\frac {4 b^2 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{35 d}+\frac {3 e (c+d x)^2 \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{7 d}+\frac {6 b e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{35 d} \]
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Rubi [A] time = 0.29, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 11, 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, 3353, 3352, 3351} \[ -\frac {8 \sqrt {2 \pi } b^{7/2} e \sin (a) \sqrt [3]{e (c+d x)} \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b}}{\sqrt [3]{c+d x}}\right )}{35 d \sqrt [3]{c+d x}}-\frac {8 \sqrt {2 \pi } b^{7/2} e \cos (a) \sqrt [3]{e (c+d x)} S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{35 d \sqrt [3]{c+d x}}-\frac {4 b^2 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{35 d}-\frac {8 b^3 e \sqrt [3]{e (c+d x)} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{35 d}+\frac {3 e (c+d x)^2 \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{7 d}+\frac {6 b e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{35 d} \]
Antiderivative was successfully verified.
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Rule 3351
Rule 3352
Rule 3353
Rule 3387
Rule 3388
Rule 3409
Rule 3415
Rule 3417
Rule 3435
Rubi steps
\begin {align*} \int (c e+d e x)^{4/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx &=\frac {\operatorname {Subst}\left (\int (e x)^{4/3} \sin \left (a+\frac {b}{x^{2/3}}\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {\left (e \sqrt [3]{e (c+d x)}\right ) \operatorname {Subst}\left (\int x^{4/3} \sin \left (a+\frac {b}{x^{2/3}}\right ) \, dx,x,c+d x\right )}{d \sqrt [3]{c+d x}}\\ &=\frac {\left (3 e \sqrt [3]{e (c+d x)}\right ) \operatorname {Subst}\left (\int x^6 \sin \left (a+\frac {b}{x^2}\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d \sqrt [3]{c+d x}}\\ &=-\frac {\left (3 e \sqrt [3]{e (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {\sin \left (a+b x^2\right )}{x^8} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d \sqrt [3]{c+d x}}\\ &=\frac {3 e (c+d x)^2 \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{7 d}-\frac {\left (6 b e \sqrt [3]{e (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {\cos \left (a+b x^2\right )}{x^6} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{7 d \sqrt [3]{c+d x}}\\ &=\frac {6 b e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{35 d}+\frac {3 e (c+d x)^2 \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{7 d}+\frac {\left (12 b^2 e \sqrt [3]{e (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {\sin \left (a+b x^2\right )}{x^4} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{35 d \sqrt [3]{c+d x}}\\ &=\frac {6 b e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{35 d}-\frac {4 b^2 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{35 d}+\frac {3 e (c+d x)^2 \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{7 d}+\frac {\left (8 b^3 e \sqrt [3]{e (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {\cos \left (a+b x^2\right )}{x^2} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{35 d \sqrt [3]{c+d x}}\\ &=-\frac {8 b^3 e \sqrt [3]{e (c+d x)} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{35 d}+\frac {6 b e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{35 d}-\frac {4 b^2 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{35 d}+\frac {3 e (c+d x)^2 \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{7 d}-\frac {\left (16 b^4 e \sqrt [3]{e (c+d x)}\right ) \operatorname {Subst}\left (\int \sin \left (a+b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{35 d \sqrt [3]{c+d x}}\\ &=-\frac {8 b^3 e \sqrt [3]{e (c+d x)} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{35 d}+\frac {6 b e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{35 d}-\frac {4 b^2 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{35 d}+\frac {3 e (c+d x)^2 \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{7 d}-\frac {\left (16 b^4 e \sqrt [3]{e (c+d x)} \cos (a)\right ) \operatorname {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{35 d \sqrt [3]{c+d x}}-\frac {\left (16 b^4 e \sqrt [3]{e (c+d x)} \sin (a)\right ) \operatorname {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{35 d \sqrt [3]{c+d x}}\\ &=-\frac {8 b^3 e \sqrt [3]{e (c+d x)} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{35 d}+\frac {6 b e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{35 d}-\frac {8 b^{7/2} e \sqrt {2 \pi } \sqrt [3]{e (c+d x)} \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{35 d \sqrt [3]{c+d x}}-\frac {8 b^{7/2} e \sqrt {2 \pi } \sqrt [3]{e (c+d x)} C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{35 d \sqrt [3]{c+d x}}-\frac {4 b^2 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{35 d}+\frac {3 e (c+d x)^2 \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.90, size = 237, normalized size = 0.79 \[ \frac {(e (c+d x))^{4/3} \left (-\frac {8 \sqrt {2 \pi } b^{7/2} \left (\sin (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )+\cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )\right )}{(c+d x)^{4/3}}+\frac {\cos \left (\frac {b}{(c+d x)^{2/3}}\right ) \left (-8 b^3 \cos (a)-4 b^2 \sin (a) (c+d x)^{2/3}+6 b \cos (a) (c+d x)^{4/3}+15 \sin (a) (c+d x)^2\right )}{c+d x}+\frac {\sin \left (\frac {b}{(c+d x)^{2/3}}\right ) \left (8 b^3 \sin (a)-4 b^2 \cos (a) (c+d x)^{2/3}-6 b \sin (a) (c+d x)^{4/3}+15 \cos (a) (c+d x)^2\right )}{c+d x}\right )}{35 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d e x + c e\right )}^{\frac {4}{3}} \sin \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {1}{3}} b}{d x + c}\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \left (d e x +c e \right )^{\frac {4}{3}} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 2.54, size = 1119, normalized size = 3.74 \[ \text {result too large to display} \]
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 \[ \int \sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{2/3}}\right )\,{\left (c\,e+d\,e\,x\right )}^{4/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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