Optimal. Leaf size=633 \[ -\frac {120960 f^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^9 d^3}+\frac {6 (d e-c f)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {720 f (d e-c f) \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac {60480 f^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}-\frac {3 (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {120 f (d e-c f) (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {5040 f^2 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {6 f (d e-c f) (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {168 f^2 (c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {3 f^2 (c+d x)^{8/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {720 f (d e-c f) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac {120960 f^2 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}+\frac {6 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {360 f (d e-c f) (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {20160 f^2 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {30 f (d e-c f) (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 f^2 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {24 f^2 (c+d x)^{7/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.44, antiderivative size = 633, normalized size of antiderivative = 1.00, number of steps
used = 20, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3512, 3377,
2718, 2717} \begin {gather*} -\frac {120960 f^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^9 d^3}-\frac {120960 f^2 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}+\frac {60480 f^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}+\frac {720 f (d e-c f) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {20160 f^2 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac {720 f \sqrt [3]{c+d x} (d e-c f) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {5040 f^2 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {360 f (c+d x)^{2/3} (d e-c f) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {1008 f^2 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {120 f (c+d x) (d e-c f) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {6 (d e-c f)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {168 f^2 (c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {30 f (c+d x)^{4/3} (d e-c f) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {6 \sqrt [3]{c+d x} (d e-c f)^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {24 f^2 (c+d x)^{7/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {6 f (c+d x)^{5/3} (d e-c f) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {3 (c+d x)^{2/3} (d e-c f)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {3 f^2 (c+d x)^{8/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2717
Rule 2718
Rule 3377
Rule 3512
Rubi steps
\begin {align*} \int (e+f x)^2 \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx &=\frac {3 \text {Subst}\left (\int \left (\frac {(d e-c f)^2 x^2 \sin (a+b x)}{d^2}+\frac {2 f (d e-c f) x^5 \sin (a+b x)}{d^2}+\frac {f^2 x^8 \sin (a+b x)}{d^2}\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d}\\ &=\frac {\left (3 f^2\right ) \text {Subst}\left (\int x^8 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}+\frac {(6 f (d e-c f)) \text {Subst}\left (\int x^5 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}+\frac {\left (3 (d e-c f)^2\right ) \text {Subst}\left (\int x^2 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}\\ &=-\frac {3 (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {6 f (d e-c f) (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {3 f^2 (c+d x)^{8/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {\left (24 f^2\right ) \text {Subst}\left (\int x^7 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^3}+\frac {(30 f (d e-c f)) \text {Subst}\left (\int x^4 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^3}+\frac {\left (6 (d e-c f)^2\right ) \text {Subst}\left (\int x \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^3}\\ &=-\frac {3 (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {6 f (d e-c f) (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {3 f^2 (c+d x)^{8/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {6 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {30 f (d e-c f) (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {24 f^2 (c+d x)^{7/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {\left (168 f^2\right ) \text {Subst}\left (\int x^6 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {(120 f (d e-c f)) \text {Subst}\left (\int x^3 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {\left (6 (d e-c f)^2\right ) \text {Subst}\left (\int \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^3}\\ &=\frac {6 (d e-c f)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {3 (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {120 f (d e-c f) (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {6 f (d e-c f) (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {168 f^2 (c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {3 f^2 (c+d x)^{8/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {6 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {30 f (d e-c f) (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {24 f^2 (c+d x)^{7/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {\left (1008 f^2\right ) \text {Subst}\left (\int x^5 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {(360 f (d e-c f)) \text {Subst}\left (\int x^2 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^3 d^3}\\ &=\frac {6 (d e-c f)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {3 (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {120 f (d e-c f) (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {6 f (d e-c f) (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {168 f^2 (c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {3 f^2 (c+d x)^{8/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {6 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {360 f (d e-c f) (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {30 f (d e-c f) (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 f^2 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {24 f^2 (c+d x)^{7/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {\left (5040 f^2\right ) \text {Subst}\left (\int x^4 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {(720 f (d e-c f)) \text {Subst}\left (\int x \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^4 d^3}\\ &=\frac {6 (d e-c f)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {720 f (d e-c f) \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {3 (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {120 f (d e-c f) (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {5040 f^2 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {6 f (d e-c f) (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {168 f^2 (c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {3 f^2 (c+d x)^{8/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {6 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {360 f (d e-c f) (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {30 f (d e-c f) (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 f^2 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {24 f^2 (c+d x)^{7/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {\left (20160 f^2\right ) \text {Subst}\left (\int x^3 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac {(720 f (d e-c f)) \text {Subst}\left (\int \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^5 d^3}\\ &=\frac {6 (d e-c f)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {720 f (d e-c f) \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {3 (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {120 f (d e-c f) (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {5040 f^2 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {6 f (d e-c f) (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {168 f^2 (c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {3 f^2 (c+d x)^{8/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {720 f (d e-c f) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {6 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {360 f (d e-c f) (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {20160 f^2 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {30 f (d e-c f) (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 f^2 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {24 f^2 (c+d x)^{7/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {\left (60480 f^2\right ) \text {Subst}\left (\int x^2 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^6 d^3}\\ &=\frac {6 (d e-c f)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {720 f (d e-c f) \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac {60480 f^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}-\frac {3 (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {120 f (d e-c f) (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {5040 f^2 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {6 f (d e-c f) (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {168 f^2 (c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {3 f^2 (c+d x)^{8/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {720 f (d e-c f) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {6 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {360 f (d e-c f) (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {20160 f^2 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {30 f (d e-c f) (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 f^2 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {24 f^2 (c+d x)^{7/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {\left (120960 f^2\right ) \text {Subst}\left (\int x \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^7 d^3}\\ &=\frac {6 (d e-c f)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {720 f (d e-c f) \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac {60480 f^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}-\frac {3 (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {120 f (d e-c f) (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {5040 f^2 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {6 f (d e-c f) (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {168 f^2 (c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {3 f^2 (c+d x)^{8/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {720 f (d e-c f) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac {120960 f^2 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}+\frac {6 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {360 f (d e-c f) (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {20160 f^2 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {30 f (d e-c f) (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 f^2 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {24 f^2 (c+d x)^{7/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {\left (120960 f^2\right ) \text {Subst}\left (\int \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^8 d^3}\\ &=-\frac {120960 f^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^9 d^3}+\frac {6 (d e-c f)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {720 f (d e-c f) \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac {60480 f^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}-\frac {3 (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {120 f (d e-c f) (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {5040 f^2 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {6 f (d e-c f) (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {168 f^2 (c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {3 f^2 (c+d x)^{8/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {720 f (d e-c f) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac {120960 f^2 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}+\frac {6 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {360 f (d e-c f) (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {20160 f^2 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {30 f (d e-c f) (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 f^2 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {24 f^2 (c+d x)^{7/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}\\ \end {align*}
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Mathematica [A]
time = 1.52, size = 256, normalized size = 0.40 \begin {gather*} \frac {-3 \left (40320 f^2-20160 b^2 f^2 (c+d x)^{2/3}+b^8 d^2 (c+d x)^{2/3} (e+f x)^2+240 b^4 f \sqrt [3]{c+d x} (6 c f+d (e+7 f x))-2 b^6 \left (9 c^2 f^2+18 c d f (e+2 f x)+d^2 \left (e^2+20 e f x+28 f^2 x^2\right )\right )\right ) \cos \left (a+b \sqrt [3]{c+d x}\right )+6 b \left (-20160 f^2 \sqrt [3]{c+d x}-12 b^4 f (c+d x)^{2/3} (5 d e+9 c f+14 d f x)+b^6 d \sqrt [3]{c+d x} (e+f x) (3 c f+d (e+4 f x))+120 b^2 f (27 c f+d (e+28 f x))\right ) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^9 d^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2703\) vs.
\(2(573)=1146\).
time = 0.06, size = 2704, normalized size = 4.27
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(2704\) |
default | \(\text {Expression too large to display}\) | \(2704\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2157 vs.
\(2 (582) = 1164\).
time = 0.43, size = 2157, normalized size = 3.41 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 336, normalized size = 0.53 \begin {gather*} \frac {3 \, {\left ({\left (56 \, b^{6} d^{2} f^{2} x^{2} + 72 \, b^{6} c d f^{2} x + 2 \, b^{6} d^{2} e^{2} + 18 \, {\left (b^{6} c^{2} - 2240\right )} f^{2} + 4 \, {\left (10 \, b^{6} d^{2} f x + 9 \, b^{6} c d f\right )} e - {\left (b^{8} d^{2} f^{2} x^{2} + 2 \, b^{8} d^{2} f x e + b^{8} d^{2} e^{2} - 20160 \, b^{2} f^{2}\right )} {\left (d x + c\right )}^{\frac {2}{3}} - 240 \, {\left (7 \, b^{4} d f^{2} x + 6 \, b^{4} c f^{2} + b^{4} d f e\right )} {\left (d x + c\right )}^{\frac {1}{3}}\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + 2 \, {\left (3360 \, b^{3} d f^{2} x + 3240 \, b^{3} c f^{2} + 120 \, b^{3} d f e - 12 \, {\left (14 \, b^{5} d f^{2} x + 9 \, b^{5} c f^{2} + 5 \, b^{5} d f e\right )} {\left (d x + c\right )}^{\frac {2}{3}} + {\left (4 \, b^{7} d^{2} f^{2} x^{2} + 3 \, b^{7} c d f^{2} x + b^{7} d^{2} e^{2} - 20160 \, b f^{2} + {\left (5 \, b^{7} d^{2} f x + 3 \, b^{7} c d f\right )} e\right )} {\left (d x + c\right )}^{\frac {1}{3}}\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{9} d^{3}} \end {gather*}
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 \begin {gather*} \int \left (e + f x\right )^{2} \sin {\left (a + b \sqrt [3]{c + d x} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1558 vs.
\(2 (582) = 1164\).
time = 5.08, size = 1558, normalized size = 2.46 \begin {gather*} \text {Too large to display} \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+b\,{\left (c+d\,x\right )}^{1/3}\right )\,{\left (e+f\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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