Optimal. Leaf size=513 \[ \frac {3 \sqrt {\frac {\pi }{2}} \cos (a) (d e-c f)^2 C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d^3}-\frac {3 \sqrt {\frac {\pi }{2}} \sin (a) (d e-c f)^2 S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d^3}+\frac {315 \sqrt {\frac {\pi }{2}} f^2 \sin (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{16 b^{9/2} d^3}+\frac {315 \sqrt {\frac {\pi }{2}} f^2 \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{16 b^{9/2} d^3}-\frac {315 f^2 \sqrt [3]{c+d x} \sin \left (a+b (c+d x)^{2/3}\right )}{16 b^4 d^3}+\frac {6 f (d e-c f) \cos \left (a+b (c+d x)^{2/3}\right )}{b^3 d^3}+\frac {105 f^2 (c+d x) \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d^3}+\frac {6 f (c+d x)^{2/3} (d e-c f) \sin \left (a+b (c+d x)^{2/3}\right )}{b^2 d^3}+\frac {21 f^2 (c+d x)^{5/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d^3}-\frac {3 f (c+d x)^{4/3} (d e-c f) \cos \left (a+b (c+d x)^{2/3}\right )}{b d^3}-\frac {3 \sqrt [3]{c+d x} (d e-c f)^2 \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}-\frac {3 f^2 (c+d x)^{7/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3} \]
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Rubi [A] time = 0.54, antiderivative size = 513, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {3433, 3385, 3354, 3352, 3351, 3379, 3296, 2638, 3386, 3353} \[ \frac {3 \sqrt {\frac {\pi }{2}} \cos (a) (d e-c f)^2 \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d^3}-\frac {3 \sqrt {\frac {\pi }{2}} \sin (a) (d e-c f)^2 S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d^3}+\frac {6 f (c+d x)^{2/3} (d e-c f) \sin \left (a+b (c+d x)^{2/3}\right )}{b^2 d^3}+\frac {6 f (d e-c f) \cos \left (a+b (c+d x)^{2/3}\right )}{b^3 d^3}+\frac {315 \sqrt {\frac {\pi }{2}} f^2 \sin (a) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt [3]{c+d x}\right )}{16 b^{9/2} d^3}+\frac {315 \sqrt {\frac {\pi }{2}} f^2 \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{16 b^{9/2} d^3}+\frac {21 f^2 (c+d x)^{5/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d^3}-\frac {315 f^2 \sqrt [3]{c+d x} \sin \left (a+b (c+d x)^{2/3}\right )}{16 b^4 d^3}+\frac {105 f^2 (c+d x) \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d^3}-\frac {3 f (c+d x)^{4/3} (d e-c f) \cos \left (a+b (c+d x)^{2/3}\right )}{b d^3}-\frac {3 \sqrt [3]{c+d x} (d e-c f)^2 \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}-\frac {3 f^2 (c+d x)^{7/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 3351
Rule 3352
Rule 3353
Rule 3354
Rule 3379
Rule 3385
Rule 3386
Rule 3433
Rubi steps
\begin {align*} \int (e+f x)^2 \sin \left (a+b (c+d x)^{2/3}\right ) \, dx &=\frac {3 \operatorname {Subst}\left (\int \left ((d e-c f)^2 x^2 \sin \left (a+b x^2\right )-2 f (-d e+c f) x^5 \sin \left (a+b x^2\right )+f^2 x^8 \sin \left (a+b x^2\right )\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}\\ &=\frac {\left (3 f^2\right ) \operatorname {Subst}\left (\int x^8 \sin \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}+\frac {(6 f (d e-c f)) \operatorname {Subst}\left (\int x^5 \sin \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}+\frac {\left (3 (d e-c f)^2\right ) \operatorname {Subst}\left (\int x^2 \sin \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}\\ &=-\frac {3 (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}-\frac {3 f^2 (c+d x)^{7/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}+\frac {\left (21 f^2\right ) \operatorname {Subst}\left (\int x^6 \cos \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{2 b d^3}+\frac {(3 f (d e-c f)) \operatorname {Subst}\left (\int x^2 \sin (a+b x) \, dx,x,(c+d x)^{2/3}\right )}{d^3}+\frac {\left (3 (d e-c f)^2\right ) \operatorname {Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{2 b d^3}\\ &=-\frac {3 (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}-\frac {3 f (d e-c f) (c+d x)^{4/3} \cos \left (a+b (c+d x)^{2/3}\right )}{b d^3}-\frac {3 f^2 (c+d x)^{7/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}+\frac {21 f^2 (c+d x)^{5/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d^3}-\frac {\left (105 f^2\right ) \operatorname {Subst}\left (\int x^4 \sin \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{4 b^2 d^3}+\frac {(6 f (d e-c f)) \operatorname {Subst}\left (\int x \cos (a+b x) \, dx,x,(c+d x)^{2/3}\right )}{b d^3}+\frac {\left (3 (d e-c f)^2 \cos (a)\right ) \operatorname {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{2 b d^3}-\frac {\left (3 (d e-c f)^2 \sin (a)\right ) \operatorname {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{2 b d^3}\\ &=-\frac {3 (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}+\frac {105 f^2 (c+d x) \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d^3}-\frac {3 f (d e-c f) (c+d x)^{4/3} \cos \left (a+b (c+d x)^{2/3}\right )}{b d^3}-\frac {3 f^2 (c+d x)^{7/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}+\frac {3 (d e-c f)^2 \sqrt {\frac {\pi }{2}} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d^3}-\frac {3 (d e-c f)^2 \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{2 b^{3/2} d^3}+\frac {6 f (d e-c f) (c+d x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{b^2 d^3}+\frac {21 f^2 (c+d x)^{5/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d^3}-\frac {\left (315 f^2\right ) \operatorname {Subst}\left (\int x^2 \cos \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{8 b^3 d^3}-\frac {(6 f (d e-c f)) \operatorname {Subst}\left (\int \sin (a+b x) \, dx,x,(c+d x)^{2/3}\right )}{b^2 d^3}\\ &=\frac {6 f (d e-c f) \cos \left (a+b (c+d x)^{2/3}\right )}{b^3 d^3}-\frac {3 (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}+\frac {105 f^2 (c+d x) \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d^3}-\frac {3 f (d e-c f) (c+d x)^{4/3} \cos \left (a+b (c+d x)^{2/3}\right )}{b d^3}-\frac {3 f^2 (c+d x)^{7/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}+\frac {3 (d e-c f)^2 \sqrt {\frac {\pi }{2}} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d^3}-\frac {3 (d e-c f)^2 \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{2 b^{3/2} d^3}-\frac {315 f^2 \sqrt [3]{c+d x} \sin \left (a+b (c+d x)^{2/3}\right )}{16 b^4 d^3}+\frac {6 f (d e-c f) (c+d x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{b^2 d^3}+\frac {21 f^2 (c+d x)^{5/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d^3}+\frac {\left (315 f^2\right ) \operatorname {Subst}\left (\int \sin \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{16 b^4 d^3}\\ &=\frac {6 f (d e-c f) \cos \left (a+b (c+d x)^{2/3}\right )}{b^3 d^3}-\frac {3 (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}+\frac {105 f^2 (c+d x) \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d^3}-\frac {3 f (d e-c f) (c+d x)^{4/3} \cos \left (a+b (c+d x)^{2/3}\right )}{b d^3}-\frac {3 f^2 (c+d x)^{7/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}+\frac {3 (d e-c f)^2 \sqrt {\frac {\pi }{2}} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d^3}-\frac {3 (d e-c f)^2 \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{2 b^{3/2} d^3}-\frac {315 f^2 \sqrt [3]{c+d x} \sin \left (a+b (c+d x)^{2/3}\right )}{16 b^4 d^3}+\frac {6 f (d e-c f) (c+d x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{b^2 d^3}+\frac {21 f^2 (c+d x)^{5/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d^3}+\frac {\left (315 f^2 \cos (a)\right ) \operatorname {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{16 b^4 d^3}+\frac {\left (315 f^2 \sin (a)\right ) \operatorname {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{16 b^4 d^3}\\ &=\frac {6 f (d e-c f) \cos \left (a+b (c+d x)^{2/3}\right )}{b^3 d^3}-\frac {3 (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}+\frac {105 f^2 (c+d x) \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d^3}-\frac {3 f (d e-c f) (c+d x)^{4/3} \cos \left (a+b (c+d x)^{2/3}\right )}{b d^3}-\frac {3 f^2 (c+d x)^{7/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}+\frac {3 (d e-c f)^2 \sqrt {\frac {\pi }{2}} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d^3}+\frac {315 f^2 \sqrt {\frac {\pi }{2}} \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{16 b^{9/2} d^3}+\frac {315 f^2 \sqrt {\frac {\pi }{2}} C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{16 b^{9/2} d^3}-\frac {3 (d e-c f)^2 \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{2 b^{3/2} d^3}-\frac {315 f^2 \sqrt [3]{c+d x} \sin \left (a+b (c+d x)^{2/3}\right )}{16 b^4 d^3}+\frac {6 f (d e-c f) (c+d x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{b^2 d^3}+\frac {21 f^2 (c+d x)^{5/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d^3}\\ \end {align*}
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Mathematica [C] time = 2.41, size = 432, normalized size = 0.84 \[ -\frac {3 i \left (\left (\cos \left (a+b (c+d x)^{2/3}\right )-i \sin \left (a+b (c+d x)^{2/3}\right )\right ) \left ((1+i) \sqrt {\frac {\pi }{2}} \left (8 b^3 (d e-c f)^2+105 i f^2\right ) \text {erf}\left (\frac {(1+i) \sqrt {b} \sqrt [3]{c+d x}}{\sqrt {2}}\right ) \left (\cos \left (b (c+d x)^{2/3}\right )+i \sin \left (b (c+d x)^{2/3}\right )\right )+2 \sqrt {b} \left (-8 i b^3 d^2 \sqrt [3]{c+d x} (e+f x)^2+4 b^2 f (c+d x)^{2/3} (c f-8 d e-7 d f x)+2 i b f (19 c f+16 d e+35 d f x)+105 f^2 \sqrt [3]{c+d x}\right )\right )+(\cos (a)+i \sin (a)) \left ((1+i) \sqrt {\frac {\pi }{2}} \left (8 b^3 (d e-c f)^2-105 i f^2\right ) \text {erfi}\left (\frac {(1+i) \sqrt {b} \sqrt [3]{c+d x}}{\sqrt {2}}\right )+2 \sqrt {b} \left (-8 i b^3 d^2 \sqrt [3]{c+d x} (e+f x)^2+4 b^2 f (c+d x)^{2/3} (-c f+8 d e+7 d f x)+2 i b f (19 c f+16 d e+35 d f x)-105 f^2 \sqrt [3]{c+d x}\right ) \left (\cos \left (b (c+d x)^{2/3}\right )+i \sin \left (b (c+d x)^{2/3}\right )\right )\right )\right )}{64 b^{9/2} d^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 308, normalized size = 0.60 \[ \frac {3 \, {\left (\sqrt {2} {\left (105 \, \pi f^{2} \sin \relax (a) + 8 \, \pi {\left (b^{3} d^{2} e^{2} - 2 \, b^{3} c d e f + b^{3} c^{2} f^{2}\right )} \cos \relax (a)\right )} \sqrt {\frac {b}{\pi }} \operatorname {C}\left (\sqrt {2} {\left (d x + c\right )}^{\frac {1}{3}} \sqrt {\frac {b}{\pi }}\right ) + \sqrt {2} {\left (105 \, \pi f^{2} \cos \relax (a) - 8 \, \pi {\left (b^{3} d^{2} e^{2} - 2 \, b^{3} c d e f + b^{3} c^{2} f^{2}\right )} \sin \relax (a)\right )} \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\sqrt {2} {\left (d x + c\right )}^{\frac {1}{3}} \sqrt {\frac {b}{\pi }}\right ) + 4 \, {\left (35 \, b^{2} d f^{2} x + 16 \, b^{2} d e f + 19 \, b^{2} c f^{2} - 4 \, {\left (b^{4} d^{2} f^{2} x^{2} + 2 \, b^{4} d^{2} e f x + b^{4} d^{2} e^{2}\right )} {\left (d x + c\right )}^{\frac {1}{3}}\right )} \cos \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right ) - 2 \, {\left (105 \, {\left (d x + c\right )}^{\frac {1}{3}} b f^{2} - 4 \, {\left (7 \, b^{3} d f^{2} x + 8 \, b^{3} d e f - b^{3} c f^{2}\right )} {\left (d x + c\right )}^{\frac {2}{3}}\right )} \sin \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )\right )}}{32 \, b^{5} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.83, size = 777, normalized size = 1.51 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 395, normalized size = 0.77 \[ \frac {-\frac {3 f^{2} \left (d x +c \right )^{\frac {7}{3}} \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {21 f^{2} \left (\frac {\left (d x +c \right )^{\frac {5}{3}} \sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}-\frac {5 \left (-\frac {\left (d x +c \right ) \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {\frac {3 \left (d x +c \right )^{\frac {1}{3}} \sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{4 b}-\frac {3 \sqrt {2}\, \sqrt {\pi }\, \left (\cos \relax (a ) \mathrm {S}\left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )+\sin \relax (a ) \FresnelC \left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{8 b^{\frac {3}{2}}}}{b}\right )}{2 b}\right )}{2 b}-\frac {3 \left (-2 c \,f^{2}+2 d e f \right ) \left (d x +c \right )^{\frac {4}{3}} \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {6 \left (-2 c \,f^{2}+2 d e f \right ) \left (\frac {\left (d x +c \right )^{\frac {2}{3}} \sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {\cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b^{2}}\right )}{b}-\frac {3 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \left (d x +c \right )^{\frac {1}{3}} \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {3 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \sqrt {2}\, \sqrt {\pi }\, \left (\cos \relax (a ) \FresnelC \left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \relax (a ) \mathrm {S}\left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{4 b^{\frac {3}{2}}}}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.50, size = 559, normalized size = 1.09 \[ \text {result too large to display} \]
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 \[ \int \sin \left (a+b\,{\left (c+d\,x\right )}^{2/3}\right )\,{\left (e+f\,x\right )}^2 \,d x \]
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 \[ \int \left (e + f x\right )^{2} \sin {\left (a + b \left (c + d x\right )^{\frac {2}{3}} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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