Optimal. Leaf size=267 \[ -\frac {45 \sqrt {\pi } e \cos (a) \sqrt [3]{e (c+d x)} C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{8 \sqrt {2} b^{7/2} d \sqrt [3]{c+d x}}+\frac {45 \sqrt {\pi } e \sin (a) \sqrt [3]{e (c+d x)} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{8 \sqrt {2} b^{7/2} d \sqrt [3]{c+d x}}+\frac {45 e \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d}+\frac {15 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d}-\frac {3 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d} \]
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Rubi [A] time = 0.27, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {3435, 3417, 3415, 3385, 3386, 3354, 3352, 3351} \[ -\frac {45 \sqrt {\pi } e \cos (a) \sqrt [3]{e (c+d x)} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt [3]{c+d x}\right )}{8 \sqrt {2} b^{7/2} d \sqrt [3]{c+d x}}+\frac {45 \sqrt {\pi } e \sin (a) \sqrt [3]{e (c+d x)} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{8 \sqrt {2} b^{7/2} d \sqrt [3]{c+d x}}+\frac {15 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d}+\frac {45 e \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d}-\frac {3 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d} \]
Antiderivative was successfully verified.
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Rule 3351
Rule 3352
Rule 3354
Rule 3385
Rule 3386
Rule 3415
Rule 3417
Rule 3435
Rubi steps
\begin {align*} \int (c e+d e x)^{4/3} \sin \left (a+b (c+d x)^{2/3}\right ) \, dx &=\frac {\operatorname {Subst}\left (\int (e x)^{4/3} \sin \left (a+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+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+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d \sqrt [3]{c+d x}}\\ &=-\frac {3 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac {\left (15 e \sqrt [3]{e (c+d x)}\right ) \operatorname {Subst}\left (\int x^4 \cos \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{2 b d \sqrt [3]{c+d x}}\\ &=-\frac {3 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac {15 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d}-\frac {\left (45 e \sqrt [3]{e (c+d x)}\right ) \operatorname {Subst}\left (\int x^2 \sin \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{4 b^2 d \sqrt [3]{c+d x}}\\ &=\frac {45 e \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d}-\frac {3 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac {15 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d}-\frac {\left (45 e \sqrt [3]{e (c+d x)}\right ) \operatorname {Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{8 b^3 d \sqrt [3]{c+d x}}\\ &=\frac {45 e \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d}-\frac {3 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac {15 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d}-\frac {\left (45 e \sqrt [3]{e (c+d x)} \cos (a)\right ) \operatorname {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{8 b^3 d \sqrt [3]{c+d x}}+\frac {\left (45 e \sqrt [3]{e (c+d x)} \sin (a)\right ) \operatorname {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{8 b^3 d \sqrt [3]{c+d x}}\\ &=\frac {45 e \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d}-\frac {3 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}-\frac {45 e \sqrt {\pi } \sqrt [3]{e (c+d x)} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{8 \sqrt {2} b^{7/2} d \sqrt [3]{c+d x}}+\frac {45 e \sqrt {\pi } \sqrt [3]{e (c+d x)} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{8 \sqrt {2} b^{7/2} d \sqrt [3]{c+d x}}+\frac {15 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d}\\ \end {align*}
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Mathematica [A] time = 0.76, size = 175, normalized size = 0.66 \[ -\frac {3 (e (c+d x))^{4/3} \left (2 \sqrt {b} \left (\sqrt [3]{c+d x} \left (4 b^2 (c+d x)^{4/3}-15\right ) \cos \left (a+b (c+d x)^{2/3}\right )-10 b (c+d x) \sin \left (a+b (c+d x)^{2/3}\right )\right )+15 \sqrt {2 \pi } \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )-15 \sqrt {2 \pi } \sin (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )\right )}{16 b^{7/2} d (c+d x)^{4/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d e x + c e\right )}^{\frac {4}{3}} \sin \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 1.09, size = 713, normalized size = 2.67 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \left (d e x +c e \right )^{\frac {4}{3}} \sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.78, size = 386, normalized size = 1.45 \[ -\frac {{\left ({\left ({\left (-3 i \, \Gamma \left (\frac {7}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + 3 i \, \Gamma \left (\frac {7}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (3 i \, \Gamma \left (\frac {7}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) - 3 i \, \Gamma \left (\frac {7}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right ) + 3 \, {\left (\Gamma \left (\frac {7}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (\frac {7}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right ) - 3 \, {\left (\Gamma \left (\frac {7}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (\frac {7}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (-\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right )\right )} \cos \relax (a) + {\left (3 \, {\left (\Gamma \left (\frac {7}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (\frac {7}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right ) + 3 \, {\left (\Gamma \left (\frac {7}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (\frac {7}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (3 i \, \Gamma \left (\frac {7}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) - 3 i \, \Gamma \left (\frac {7}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (3 i \, \Gamma \left (\frac {7}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) - 3 i \, \Gamma \left (\frac {7}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (-\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right )\right )} \sin \relax (a)\right )} \sqrt {{\left (d x + c\right )}^{\frac {2}{3}} b} e^{\frac {4}{3}}}{8 \, {\left (d x + c\right )}^{\frac {1}{3}} b^{4} d} \]
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 \[ \int \sin \left (a+b\,{\left (c+d\,x\right )}^{2/3}\right )\,{\left (c\,e+d\,e\,x\right )}^{4/3} \,d x \]
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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