Optimal. Leaf size=135 \[ \frac{a^2 \sin (c+d x)}{d^2}-\frac{a^2 x \cos (c+d x)}{d}+\frac{4 a b x \sin (c+d x)}{d^2}+\frac{4 a b \cos (c+d x)}{d^3}-\frac{2 a b x^2 \cos (c+d x)}{d}+\frac{3 b^2 x^2 \sin (c+d x)}{d^2}-\frac{6 b^2 \sin (c+d x)}{d^4}+\frac{6 b^2 x \cos (c+d x)}{d^3}-\frac{b^2 x^3 \cos (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.186066, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {6742, 3296, 2637, 2638} \[ \frac{a^2 \sin (c+d x)}{d^2}-\frac{a^2 x \cos (c+d x)}{d}+\frac{4 a b x \sin (c+d x)}{d^2}+\frac{4 a b \cos (c+d x)}{d^3}-\frac{2 a b x^2 \cos (c+d x)}{d}+\frac{3 b^2 x^2 \sin (c+d x)}{d^2}-\frac{6 b^2 \sin (c+d x)}{d^4}+\frac{6 b^2 x \cos (c+d x)}{d^3}-\frac{b^2 x^3 \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6742
Rule 3296
Rule 2637
Rule 2638
Rubi steps
\begin{align*} \int x (a+b x)^2 \sin (c+d x) \, dx &=\int \left (a^2 x \sin (c+d x)+2 a b x^2 \sin (c+d x)+b^2 x^3 \sin (c+d x)\right ) \, dx\\ &=a^2 \int x \sin (c+d x) \, dx+(2 a b) \int x^2 \sin (c+d x) \, dx+b^2 \int x^3 \sin (c+d x) \, dx\\ &=-\frac{a^2 x \cos (c+d x)}{d}-\frac{2 a b x^2 \cos (c+d x)}{d}-\frac{b^2 x^3 \cos (c+d x)}{d}+\frac{a^2 \int \cos (c+d x) \, dx}{d}+\frac{(4 a b) \int x \cos (c+d x) \, dx}{d}+\frac{\left (3 b^2\right ) \int x^2 \cos (c+d x) \, dx}{d}\\ &=-\frac{a^2 x \cos (c+d x)}{d}-\frac{2 a b x^2 \cos (c+d x)}{d}-\frac{b^2 x^3 \cos (c+d x)}{d}+\frac{a^2 \sin (c+d x)}{d^2}+\frac{4 a b x \sin (c+d x)}{d^2}+\frac{3 b^2 x^2 \sin (c+d x)}{d^2}-\frac{(4 a b) \int \sin (c+d x) \, dx}{d^2}-\frac{\left (6 b^2\right ) \int x \sin (c+d x) \, dx}{d^2}\\ &=\frac{4 a b \cos (c+d x)}{d^3}+\frac{6 b^2 x \cos (c+d x)}{d^3}-\frac{a^2 x \cos (c+d x)}{d}-\frac{2 a b x^2 \cos (c+d x)}{d}-\frac{b^2 x^3 \cos (c+d x)}{d}+\frac{a^2 \sin (c+d x)}{d^2}+\frac{4 a b x \sin (c+d x)}{d^2}+\frac{3 b^2 x^2 \sin (c+d x)}{d^2}-\frac{\left (6 b^2\right ) \int \cos (c+d x) \, dx}{d^3}\\ &=\frac{4 a b \cos (c+d x)}{d^3}+\frac{6 b^2 x \cos (c+d x)}{d^3}-\frac{a^2 x \cos (c+d x)}{d}-\frac{2 a b x^2 \cos (c+d x)}{d}-\frac{b^2 x^3 \cos (c+d x)}{d}-\frac{6 b^2 \sin (c+d x)}{d^4}+\frac{a^2 \sin (c+d x)}{d^2}+\frac{4 a b x \sin (c+d x)}{d^2}+\frac{3 b^2 x^2 \sin (c+d x)}{d^2}\\ \end{align*}
Mathematica [A] time = 0.211745, size = 87, normalized size = 0.64 \[ \frac{\left (a^2 d^2+4 a b d^2 x+3 b^2 \left (d^2 x^2-2\right )\right ) \sin (c+d x)-d \left (a^2 d^2 x+2 a b \left (d^2 x^2-2\right )+b^2 x \left (d^2 x^2-6\right )\right ) \cos (c+d x)}{d^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.008, size = 281, normalized size = 2.1 \begin{align*}{\frac{1}{{d}^{2}} \left ({\frac{{b}^{2} \left ( - \left ( dx+c \right ) ^{3}\cos \left ( dx+c \right ) +3\, \left ( dx+c \right ) ^{2}\sin \left ( dx+c \right ) -6\,\sin \left ( dx+c \right ) +6\, \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) }{{d}^{2}}}+2\,{\frac{ab \left ( - \left ( dx+c \right ) ^{2}\cos \left ( dx+c \right ) +2\,\cos \left ( dx+c \right ) +2\, \left ( dx+c \right ) \sin \left ( dx+c \right ) \right ) }{d}}-3\,{\frac{{b}^{2}c \left ( - \left ( dx+c \right ) ^{2}\cos \left ( dx+c \right ) +2\,\cos \left ( dx+c \right ) +2\, \left ( dx+c \right ) \sin \left ( dx+c \right ) \right ) }{{d}^{2}}}+{a}^{2} \left ( \sin \left ( dx+c \right ) - \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) -4\,{\frac{abc \left ( \sin \left ( dx+c \right ) - \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) }{d}}+3\,{\frac{{b}^{2}{c}^{2} \left ( \sin \left ( dx+c \right ) - \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) }{{d}^{2}}}+{a}^{2}c\cos \left ( dx+c \right ) -2\,{\frac{ab{c}^{2}\cos \left ( dx+c \right ) }{d}}+{\frac{{b}^{2}{c}^{3}\cos \left ( dx+c \right ) }{{d}^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.03814, size = 350, normalized size = 2.59 \begin{align*} \frac{a^{2} c \cos \left (d x + c\right ) + \frac{b^{2} c^{3} \cos \left (d x + c\right )}{d^{2}} - \frac{2 \, a b c^{2} \cos \left (d x + c\right )}{d} -{\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a^{2} - \frac{3 \,{\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} b^{2} c^{2}}{d^{2}} + \frac{4 \,{\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a b c}{d} + \frac{3 \,{\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \,{\left (d x + c\right )} \sin \left (d x + c\right )\right )} b^{2} c}{d^{2}} - \frac{2 \,{\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \,{\left (d x + c\right )} \sin \left (d x + c\right )\right )} a b}{d} - \frac{{\left ({\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \cos \left (d x + c\right ) - 3 \,{\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} b^{2}}{d^{2}}}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.65226, size = 200, normalized size = 1.48 \begin{align*} -\frac{{\left (b^{2} d^{3} x^{3} + 2 \, a b d^{3} x^{2} - 4 \, a b d +{\left (a^{2} d^{3} - 6 \, b^{2} d\right )} x\right )} \cos \left (d x + c\right ) -{\left (3 \, b^{2} d^{2} x^{2} + 4 \, a b d^{2} x + a^{2} d^{2} - 6 \, b^{2}\right )} \sin \left (d x + c\right )}{d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.28016, size = 172, normalized size = 1.27 \begin{align*} \begin{cases} - \frac{a^{2} x \cos{\left (c + d x \right )}}{d} + \frac{a^{2} \sin{\left (c + d x \right )}}{d^{2}} - \frac{2 a b x^{2} \cos{\left (c + d x \right )}}{d} + \frac{4 a b x \sin{\left (c + d x \right )}}{d^{2}} + \frac{4 a b \cos{\left (c + d x \right )}}{d^{3}} - \frac{b^{2} x^{3} \cos{\left (c + d x \right )}}{d} + \frac{3 b^{2} x^{2} \sin{\left (c + d x \right )}}{d^{2}} + \frac{6 b^{2} x \cos{\left (c + d x \right )}}{d^{3}} - \frac{6 b^{2} \sin{\left (c + d x \right )}}{d^{4}} & \text{for}\: d \neq 0 \\\left (\frac{a^{2} x^{2}}{2} + \frac{2 a b x^{3}}{3} + \frac{b^{2} x^{4}}{4}\right ) \sin{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.10855, size = 128, normalized size = 0.95 \begin{align*} -\frac{{\left (b^{2} d^{3} x^{3} + 2 \, a b d^{3} x^{2} + a^{2} d^{3} x - 6 \, b^{2} d x - 4 \, a b d\right )} \cos \left (d x + c\right )}{d^{4}} + \frac{{\left (3 \, b^{2} d^{2} x^{2} + 4 \, a b d^{2} x + a^{2} d^{2} - 6 \, b^{2}\right )} \sin \left (d x + c\right )}{d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]