Optimal. Leaf size=299 \[ -\frac {8 b^3 e \sqrt [3]{e (c+d x)} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{35 d}+\frac {6 b e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{35 d}-\frac {8 b^{7/2} e \sqrt {2 \pi } \sqrt [3]{e (c+d x)} \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{35 d \sqrt [3]{c+d x}}-\frac {8 b^{7/2} e \sqrt {2 \pi } \sqrt [3]{e (c+d x)} C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{35 d \sqrt [3]{c+d x}}-\frac {4 b^2 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{35 d}+\frac {3 e (c+d x)^2 \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{7 d} \]
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Rubi [A]
time = 0.22, 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 = {3516, 3498,
3496, 3490, 3468, 3469, 3434, 3433, 3432} \begin {gather*} -\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 {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} \end {gather*}
Antiderivative was successfully verified.
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Rule 3432
Rule 3433
Rule 3434
Rule 3468
Rule 3469
Rule 3490
Rule 3496
Rule 3498
Rule 3516
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 {\text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 0.61, size = 237, normalized size = 0.79 \begin {gather*} \frac {(e (c+d x))^{4/3} \left (\frac {\cos \left (\frac {b}{(c+d x)^{2/3}}\right ) \left (-8 b^3 \cos (a)+6 b (c+d x)^{4/3} \cos (a)-4 b^2 (c+d x)^{2/3} \sin (a)+15 (c+d x)^2 \sin (a)\right )}{c+d x}-\frac {8 b^{7/2} \sqrt {2 \pi } \left (\cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )+C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)\right )}{(c+d x)^{4/3}}+\frac {\left (-4 b^2 (c+d x)^{2/3} \cos (a)+15 (c+d x)^2 \cos (a)+8 b^3 \sin (a)-6 b (c+d x)^{4/3} \sin (a)\right ) \sin \left (\frac {b}{(c+d x)^{2/3}}\right )}{c+d x}\right )}{35 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (d e x +c e \right )^{\frac {4}{3}} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 1.08, size = 1231, normalized size = 4.12 \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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \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 )}^{4/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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