Optimal. Leaf size=630 \[ \frac {2 \sqrt {2 \pi } b^{3/2} \sin (a) (d e-c f)^2 C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac {2 \sqrt {2 \pi } b^{3/2} \cos (a) (d e-c f)^2 S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac {16 \sqrt {2 \pi } b^{9/2} f^2 \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{315 d^3}+\frac {16 \sqrt {2 \pi } b^{9/2} f^2 \sin (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{315 d^3}+\frac {16 b^4 f^2 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{315 d^3}+\frac {b^3 f \cos (a) (d e-c f) \text {Ci}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac {b^3 f \sin (a) (d e-c f) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac {8 b^3 f^2 (c+d x) \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{315 d^3}-\frac {b^2 f (c+d x)^{2/3} (d e-c f) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac {4 b^2 f^2 (c+d x)^{5/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{105 d^3}+\frac {f (c+d x)^2 (d e-c f) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {(c+d x) (d e-c f)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {b f (c+d x)^{4/3} (d e-c f) \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {2 b \sqrt [3]{c+d x} (d e-c f)^2 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{3 d^3}+\frac {2 b f^2 (c+d x)^{7/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{21 d^3} \]
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Rubi [A] time = 0.75, antiderivative size = 630, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 13, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {3433, 3409, 3387, 3388, 3353, 3352, 3351, 3379, 3297, 3303, 3299, 3302, 3354} \[ \frac {b^3 f \cos (a) (d e-c f) \text {CosIntegral}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {2 \sqrt {2 \pi } b^{3/2} \sin (a) (d e-c f)^2 \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b}}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac {2 \sqrt {2 \pi } b^{3/2} \cos (a) (d e-c f)^2 S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac {b^3 f \sin (a) (d e-c f) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac {b^2 f (c+d x)^{2/3} (d e-c f) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac {16 \sqrt {2 \pi } b^{9/2} f^2 \cos (a) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b}}{\sqrt [3]{c+d x}}\right )}{315 d^3}+\frac {16 \sqrt {2 \pi } b^{9/2} f^2 \sin (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{315 d^3}+\frac {16 b^4 f^2 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{315 d^3}-\frac {4 b^2 f^2 (c+d x)^{5/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{105 d^3}-\frac {8 b^3 f^2 (c+d x) \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{315 d^3}+\frac {f (c+d x)^2 (d e-c f) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {(c+d x) (d e-c f)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {b f (c+d x)^{4/3} (d e-c f) \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {2 b \sqrt [3]{c+d x} (d e-c f)^2 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{3 d^3}+\frac {2 b f^2 (c+d x)^{7/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{21 d^3} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3351
Rule 3352
Rule 3353
Rule 3354
Rule 3379
Rule 3387
Rule 3388
Rule 3409
Rule 3433
Rubi steps
\begin {align*} \int (e+f x)^2 \sin \left (a+\frac {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+\frac {b}{x^2}\right )-2 f (-d e+c f) x^5 \sin \left (a+\frac {b}{x^2}\right )+f^2 x^8 \sin \left (a+\frac {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+\frac {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+\frac {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+\frac {b}{x^2}\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}\\ &=-\frac {\left (3 f^2\right ) \operatorname {Subst}\left (\int \frac {\sin \left (a+b x^2\right )}{x^{10}} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac {(3 f (d e-c f)) \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{x^4} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{d^3}-\frac {\left (3 (d e-c f)^2\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 )}{d^3}\\ &=\frac {(d e-c f)^2 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {f (d e-c f) (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{3 d^3}-\frac {\left (2 b f^2\right ) \operatorname {Subst}\left (\int \frac {\cos \left (a+b x^2\right )}{x^8} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{3 d^3}-\frac {(b f (d e-c f)) \operatorname {Subst}\left (\int \frac {\cos (a+b x)}{x^3} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{d^3}-\frac {\left (2 b (d e-c f)^2\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 )}{d^3}\\ &=\frac {2 b (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {b f (d e-c f) (c+d x)^{4/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {2 b f^2 (c+d x)^{7/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{21 d^3}+\frac {(d e-c f)^2 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {f (d e-c f) (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{3 d^3}+\frac {\left (4 b^2 f^2\right ) \operatorname {Subst}\left (\int \frac {\sin \left (a+b x^2\right )}{x^6} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{21 d^3}+\frac {\left (b^2 f (d e-c f)\right ) \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{x^2} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {\left (4 b^2 (d e-c f)^2\right ) \operatorname {Subst}\left (\int \sin \left (a+b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d^3}\\ &=\frac {2 b (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {b f (d e-c f) (c+d x)^{4/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {2 b f^2 (c+d x)^{7/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{21 d^3}-\frac {b^2 f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {(d e-c f)^2 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}-\frac {4 b^2 f^2 (c+d x)^{5/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{105 d^3}+\frac {f (d e-c f) (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{3 d^3}+\frac {\left (8 b^3 f^2\right ) \operatorname {Subst}\left (\int \frac {\cos \left (a+b x^2\right )}{x^4} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{105 d^3}+\frac {\left (b^3 f (d e-c f)\right ) \operatorname {Subst}\left (\int \frac {\cos (a+b x)}{x} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {\left (4 b^2 (d e-c f)^2 \cos (a)\right ) \operatorname {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac {\left (4 b^2 (d e-c f)^2 \sin (a)\right ) \operatorname {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d^3}\\ &=\frac {2 b (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}-\frac {8 b^3 f^2 (c+d x) \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{315 d^3}+\frac {b f (d e-c f) (c+d x)^{4/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {2 b f^2 (c+d x)^{7/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{21 d^3}+\frac {2 b^{3/2} (d e-c f)^2 \sqrt {2 \pi } \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac {2 b^{3/2} (d e-c f)^2 \sqrt {2 \pi } C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{d^3}-\frac {b^2 f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {(d e-c f)^2 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}-\frac {4 b^2 f^2 (c+d x)^{5/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{105 d^3}+\frac {f (d e-c f) (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{3 d^3}-\frac {\left (16 b^4 f^2\right ) \operatorname {Subst}\left (\int \frac {\sin \left (a+b x^2\right )}{x^2} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{315 d^3}+\frac {\left (b^3 f (d e-c f) \cos (a)\right ) \operatorname {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac {\left (b^3 f (d e-c f) \sin (a)\right ) \operatorname {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{2 d^3}\\ &=\frac {2 b (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}-\frac {8 b^3 f^2 (c+d x) \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{315 d^3}+\frac {b f (d e-c f) (c+d x)^{4/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {2 b f^2 (c+d x)^{7/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{21 d^3}+\frac {b^3 f (d e-c f) \cos (a) \text {Ci}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {2 b^{3/2} (d e-c f)^2 \sqrt {2 \pi } \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac {2 b^{3/2} (d e-c f)^2 \sqrt {2 \pi } C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{d^3}+\frac {16 b^4 f^2 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{315 d^3}-\frac {b^2 f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {(d e-c f)^2 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}-\frac {4 b^2 f^2 (c+d x)^{5/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{105 d^3}+\frac {f (d e-c f) (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{3 d^3}-\frac {b^3 f (d e-c f) \sin (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac {\left (32 b^5 f^2\right ) \operatorname {Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{315 d^3}\\ &=\frac {2 b (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}-\frac {8 b^3 f^2 (c+d x) \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{315 d^3}+\frac {b f (d e-c f) (c+d x)^{4/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {2 b f^2 (c+d x)^{7/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{21 d^3}+\frac {b^3 f (d e-c f) \cos (a) \text {Ci}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {2 b^{3/2} (d e-c f)^2 \sqrt {2 \pi } \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac {2 b^{3/2} (d e-c f)^2 \sqrt {2 \pi } C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{d^3}+\frac {16 b^4 f^2 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{315 d^3}-\frac {b^2 f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {(d e-c f)^2 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}-\frac {4 b^2 f^2 (c+d x)^{5/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{105 d^3}+\frac {f (d e-c f) (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{3 d^3}-\frac {b^3 f (d e-c f) \sin (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac {\left (32 b^5 f^2 \cos (a)\right ) \operatorname {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{315 d^3}+\frac {\left (32 b^5 f^2 \sin (a)\right ) \operatorname {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{315 d^3}\\ &=\frac {2 b (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}-\frac {8 b^3 f^2 (c+d x) \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{315 d^3}+\frac {b f (d e-c f) (c+d x)^{4/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {2 b f^2 (c+d x)^{7/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{21 d^3}+\frac {b^3 f (d e-c f) \cos (a) \text {Ci}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac {16 b^{9/2} f^2 \sqrt {2 \pi } \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{315 d^3}+\frac {2 b^{3/2} (d e-c f)^2 \sqrt {2 \pi } \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac {2 b^{3/2} (d e-c f)^2 \sqrt {2 \pi } C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{d^3}+\frac {16 b^{9/2} f^2 \sqrt {2 \pi } S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{315 d^3}+\frac {16 b^4 f^2 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{315 d^3}-\frac {b^2 f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac {(d e-c f)^2 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}-\frac {4 b^2 f^2 (c+d x)^{5/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{105 d^3}+\frac {f (d e-c f) (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{3 d^3}-\frac {b^3 f (d e-c f) \sin (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d^3}\\ \end {align*}
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Mathematica [C] time = 3.10, size = 613, normalized size = 0.97 \[ \frac {i e^{-i a} \left (315 i e^{2 i a} b^3 f (c f-d e) \text {Ei}\left (\frac {i b}{(c+d x)^{2/3}}\right )+4 \sqrt [4]{-1} \sqrt {\pi } e^{2 i a} b^{3/2} \left (f^2 \left (8 b^3+315 i c^2\right )-630 i c d e f+315 i d^2 e^2\right ) \text {erfi}\left (\frac {\sqrt [4]{-1} \sqrt {b}}{\sqrt [3]{c+d x}}\right )-\sqrt [3]{c+d x} e^{i \left (2 a+\frac {b}{(c+d x)^{2/3}}\right )} \left (32 b^4 f^2-16 i b^3 f^2 (c+d x)^{2/3}+3 b^2 f \sqrt [3]{c+d x} (97 c f-105 d e-8 d f x)+15 i b \left (f^2 \left (67 c^2-13 c d x+4 d^2 x^2\right )+21 d e f (d x-7 c)+84 d^2 e^2\right )+210 (c+d x)^{2/3} \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )\right )+315 i b^3 f (c f-d e) \text {Ei}\left (-\frac {i b}{(c+d x)^{2/3}}\right )-4 \sqrt [4]{-1} \sqrt {\pi } b^{3/2} \left (f^2 \left (315 c^2+8 i b^3\right )-630 c d e f+315 d^2 e^2\right ) \text {erfi}\left (\frac {(-1)^{3/4} \sqrt {b}}{\sqrt [3]{c+d x}}\right )+\sqrt [3]{c+d x} e^{-\frac {i b}{(c+d x)^{2/3}}} \left (32 b^4 f^2+16 i b^3 f^2 (c+d x)^{2/3}+3 b^2 f \sqrt [3]{c+d x} (97 c f-105 d e-8 d f x)-15 i b \left (f^2 \left (67 c^2-13 c d x+4 d^2 x^2\right )+21 d e f (d x-7 c)+84 d^2 e^2\right )+210 (c+d x)^{2/3} \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )\right )\right )}{1260 d^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 494, normalized size = 0.78 \[ \frac {315 \, {\left (b^{3} d e f - b^{3} c f^{2}\right )} \cos \relax (a) \operatorname {Ci}\left (\frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + 315 \, {\left (b^{3} d e f - b^{3} c f^{2}\right )} \cos \relax (a) \operatorname {Ci}\left (-\frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) - 8 \, \sqrt {2} {\left (8 \, \pi b^{4} f^{2} \cos \relax (a) - 315 \, \pi {\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} \sin \relax (a)\right )} \sqrt {\frac {b}{\pi }} \operatorname {C}\left (\frac {\sqrt {2} \sqrt {\frac {b}{\pi }}}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + 8 \, \sqrt {2} {\left (8 \, \pi b^{4} f^{2} \sin \relax (a) + 315 \, \pi {\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} \cos \relax (a)\right )} \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\frac {\sqrt {2} \sqrt {\frac {b}{\pi }}}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - 630 \, {\left (b^{3} d e f - b^{3} c f^{2}\right )} \sin \relax (a) \operatorname {Si}\left (\frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) - 2 \, {\left (16 \, b^{3} d f^{2} x + 16 \, b^{3} c f^{2} - 15 \, {\left (4 \, b d^{2} f^{2} x^{2} + 84 \, b d^{2} e^{2} - 147 \, b c d e f + 67 \, b c^{2} f^{2} + {\left (21 \, b d^{2} e f - 13 \, b c d f^{2}\right )} x\right )} {\left (d x + c\right )}^{\frac {1}{3}}\right )} \cos \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {1}{3}} b}{d x + c}\right ) + 2 \, {\left (210 \, d^{3} f^{2} x^{3} + 630 \, d^{3} e f x^{2} + 32 \, {\left (d x + c\right )}^{\frac {1}{3}} b^{4} f^{2} + 630 \, d^{3} e^{2} x + 630 \, c d^{2} e^{2} - 630 \, c^{2} d e f + 210 \, c^{3} f^{2} - 3 \, {\left (8 \, b^{2} d f^{2} x + 105 \, b^{2} d e f - 97 \, b^{2} c f^{2}\right )} {\left (d x + c\right )}^{\frac {2}{3}}\right )} \sin \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {1}{3}} b}{d x + c}\right )}{1260 \, d^{3}} \]
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 \[ \int {\left (f x + e\right )}^{2} \sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 452, normalized size = 0.72 \[ \frac {\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \left (d x +c \right ) \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )-2 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) b \left (-\left (d x +c \right )^{\frac {1}{3}} \cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )-\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \relax (a ) \mathrm {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )^{\frac {1}{3}}}\right )+\sin \relax (a ) \FresnelC \left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )^{\frac {1}{3}}}\right )\right )\right )+\frac {\left (-2 c \,f^{2}+2 d e f \right ) \left (d x +c \right )^{2} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{2}-\left (-2 c \,f^{2}+2 d e f \right ) b \left (-\frac {\left (d x +c \right )^{\frac {4}{3}} \cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{4}-\frac {b \left (-\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right ) \left (d x +c \right )^{\frac {2}{3}}}{2}+b \left (\frac {\cos \relax (a ) \Ci \left (\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{2}-\frac {\sin \relax (a ) \Si \left (\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{2}\right )\right )}{2}\right )+\frac {f^{2} \left (d x +c \right )^{3} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{3}-\frac {2 f^{2} b \left (-\frac {\left (d x +c \right )^{\frac {7}{3}} \cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{7}-\frac {2 b \left (-\frac {\left (d x +c \right )^{\frac {5}{3}} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{5}+\frac {2 b \left (-\frac {\left (d x +c \right ) \cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{3}-\frac {2 b \left (-\left (d x +c \right )^{\frac {1}{3}} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )+\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \relax (a ) \FresnelC \left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )^{\frac {1}{3}}}\right )-\sin \relax (a ) \mathrm {S}\left (\frac {\sqrt {b}\, \sqrt {2}}{\sqrt {\pi }\, \left (d x +c \right )^{\frac {1}{3}}}\right )\right )\right )}{3}\right )}{5}\right )}{7}\right )}{3}}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.37, size = 1258, normalized size = 2.00 \[ \text {result too large to display} \]
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 \[ \int \sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{2/3}}\right )\,{\left (e+f\,x\right )}^2 \,d x \]
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 \[ \int \left (e + f x\right )^{2} \sin {\left (a + \frac {b}{\left (c + d x\right )^{\frac {2}{3}}} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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