Optimal. Leaf size=236 \[ \frac{2 a^2 x \sin (c+d x)}{d^2}+\frac{2 a^2 \cos (c+d x)}{d^3}-\frac{a^2 x^2 \cos (c+d x)}{d}+\frac{8 a b x^3 \sin (c+d x)}{d^2}+\frac{24 a b x^2 \cos (c+d x)}{d^3}-\frac{48 a b x \sin (c+d x)}{d^4}-\frac{48 a b \cos (c+d x)}{d^5}-\frac{2 a b x^4 \cos (c+d x)}{d}+\frac{6 b^2 x^5 \sin (c+d x)}{d^2}-\frac{120 b^2 x^3 \sin (c+d x)}{d^4}+\frac{30 b^2 x^4 \cos (c+d x)}{d^3}-\frac{360 b^2 x^2 \cos (c+d x)}{d^5}+\frac{720 b^2 x \sin (c+d x)}{d^6}+\frac{720 b^2 \cos (c+d x)}{d^7}-\frac{b^2 x^6 \cos (c+d x)}{d} \]
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Rubi [A] time = 0.327105, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {3339, 3296, 2638} \[ \frac{2 a^2 x \sin (c+d x)}{d^2}+\frac{2 a^2 \cos (c+d x)}{d^3}-\frac{a^2 x^2 \cos (c+d x)}{d}+\frac{8 a b x^3 \sin (c+d x)}{d^2}+\frac{24 a b x^2 \cos (c+d x)}{d^3}-\frac{48 a b x \sin (c+d x)}{d^4}-\frac{48 a b \cos (c+d x)}{d^5}-\frac{2 a b x^4 \cos (c+d x)}{d}+\frac{6 b^2 x^5 \sin (c+d x)}{d^2}-\frac{120 b^2 x^3 \sin (c+d x)}{d^4}+\frac{30 b^2 x^4 \cos (c+d x)}{d^3}-\frac{360 b^2 x^2 \cos (c+d x)}{d^5}+\frac{720 b^2 x \sin (c+d x)}{d^6}+\frac{720 b^2 \cos (c+d x)}{d^7}-\frac{b^2 x^6 \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3339
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int x^2 \left (a+b x^2\right )^2 \sin (c+d x) \, dx &=\int \left (a^2 x^2 \sin (c+d x)+2 a b x^4 \sin (c+d x)+b^2 x^6 \sin (c+d x)\right ) \, dx\\ &=a^2 \int x^2 \sin (c+d x) \, dx+(2 a b) \int x^4 \sin (c+d x) \, dx+b^2 \int x^6 \sin (c+d x) \, dx\\ &=-\frac{a^2 x^2 \cos (c+d x)}{d}-\frac{2 a b x^4 \cos (c+d x)}{d}-\frac{b^2 x^6 \cos (c+d x)}{d}+\frac{\left (2 a^2\right ) \int x \cos (c+d x) \, dx}{d}+\frac{(8 a b) \int x^3 \cos (c+d x) \, dx}{d}+\frac{\left (6 b^2\right ) \int x^5 \cos (c+d x) \, dx}{d}\\ &=-\frac{a^2 x^2 \cos (c+d x)}{d}-\frac{2 a b x^4 \cos (c+d x)}{d}-\frac{b^2 x^6 \cos (c+d x)}{d}+\frac{2 a^2 x \sin (c+d x)}{d^2}+\frac{8 a b x^3 \sin (c+d x)}{d^2}+\frac{6 b^2 x^5 \sin (c+d x)}{d^2}-\frac{\left (2 a^2\right ) \int \sin (c+d x) \, dx}{d^2}-\frac{(24 a b) \int x^2 \sin (c+d x) \, dx}{d^2}-\frac{\left (30 b^2\right ) \int x^4 \sin (c+d x) \, dx}{d^2}\\ &=\frac{2 a^2 \cos (c+d x)}{d^3}+\frac{24 a b x^2 \cos (c+d x)}{d^3}-\frac{a^2 x^2 \cos (c+d x)}{d}+\frac{30 b^2 x^4 \cos (c+d x)}{d^3}-\frac{2 a b x^4 \cos (c+d x)}{d}-\frac{b^2 x^6 \cos (c+d x)}{d}+\frac{2 a^2 x \sin (c+d x)}{d^2}+\frac{8 a b x^3 \sin (c+d x)}{d^2}+\frac{6 b^2 x^5 \sin (c+d x)}{d^2}-\frac{(48 a b) \int x \cos (c+d x) \, dx}{d^3}-\frac{\left (120 b^2\right ) \int x^3 \cos (c+d x) \, dx}{d^3}\\ &=\frac{2 a^2 \cos (c+d x)}{d^3}+\frac{24 a b x^2 \cos (c+d x)}{d^3}-\frac{a^2 x^2 \cos (c+d x)}{d}+\frac{30 b^2 x^4 \cos (c+d x)}{d^3}-\frac{2 a b x^4 \cos (c+d x)}{d}-\frac{b^2 x^6 \cos (c+d x)}{d}-\frac{48 a b x \sin (c+d x)}{d^4}+\frac{2 a^2 x \sin (c+d x)}{d^2}-\frac{120 b^2 x^3 \sin (c+d x)}{d^4}+\frac{8 a b x^3 \sin (c+d x)}{d^2}+\frac{6 b^2 x^5 \sin (c+d x)}{d^2}+\frac{(48 a b) \int \sin (c+d x) \, dx}{d^4}+\frac{\left (360 b^2\right ) \int x^2 \sin (c+d x) \, dx}{d^4}\\ &=-\frac{48 a b \cos (c+d x)}{d^5}+\frac{2 a^2 \cos (c+d x)}{d^3}-\frac{360 b^2 x^2 \cos (c+d x)}{d^5}+\frac{24 a b x^2 \cos (c+d x)}{d^3}-\frac{a^2 x^2 \cos (c+d x)}{d}+\frac{30 b^2 x^4 \cos (c+d x)}{d^3}-\frac{2 a b x^4 \cos (c+d x)}{d}-\frac{b^2 x^6 \cos (c+d x)}{d}-\frac{48 a b x \sin (c+d x)}{d^4}+\frac{2 a^2 x \sin (c+d x)}{d^2}-\frac{120 b^2 x^3 \sin (c+d x)}{d^4}+\frac{8 a b x^3 \sin (c+d x)}{d^2}+\frac{6 b^2 x^5 \sin (c+d x)}{d^2}+\frac{\left (720 b^2\right ) \int x \cos (c+d x) \, dx}{d^5}\\ &=-\frac{48 a b \cos (c+d x)}{d^5}+\frac{2 a^2 \cos (c+d x)}{d^3}-\frac{360 b^2 x^2 \cos (c+d x)}{d^5}+\frac{24 a b x^2 \cos (c+d x)}{d^3}-\frac{a^2 x^2 \cos (c+d x)}{d}+\frac{30 b^2 x^4 \cos (c+d x)}{d^3}-\frac{2 a b x^4 \cos (c+d x)}{d}-\frac{b^2 x^6 \cos (c+d x)}{d}+\frac{720 b^2 x \sin (c+d x)}{d^6}-\frac{48 a b x \sin (c+d x)}{d^4}+\frac{2 a^2 x \sin (c+d x)}{d^2}-\frac{120 b^2 x^3 \sin (c+d x)}{d^4}+\frac{8 a b x^3 \sin (c+d x)}{d^2}+\frac{6 b^2 x^5 \sin (c+d x)}{d^2}-\frac{\left (720 b^2\right ) \int \sin (c+d x) \, dx}{d^6}\\ &=\frac{720 b^2 \cos (c+d x)}{d^7}-\frac{48 a b \cos (c+d x)}{d^5}+\frac{2 a^2 \cos (c+d x)}{d^3}-\frac{360 b^2 x^2 \cos (c+d x)}{d^5}+\frac{24 a b x^2 \cos (c+d x)}{d^3}-\frac{a^2 x^2 \cos (c+d x)}{d}+\frac{30 b^2 x^4 \cos (c+d x)}{d^3}-\frac{2 a b x^4 \cos (c+d x)}{d}-\frac{b^2 x^6 \cos (c+d x)}{d}+\frac{720 b^2 x \sin (c+d x)}{d^6}-\frac{48 a b x \sin (c+d x)}{d^4}+\frac{2 a^2 x \sin (c+d x)}{d^2}-\frac{120 b^2 x^3 \sin (c+d x)}{d^4}+\frac{8 a b x^3 \sin (c+d x)}{d^2}+\frac{6 b^2 x^5 \sin (c+d x)}{d^2}\\ \end{align*}
Mathematica [A] time = 0.393481, size = 139, normalized size = 0.59 \[ \frac{2 d x \left (a^2 d^4+4 a b d^2 \left (d^2 x^2-6\right )+3 b^2 \left (d^4 x^4-20 d^2 x^2+120\right )\right ) \sin (c+d x)-\left (a^2 d^4 \left (d^2 x^2-2\right )+2 a b d^2 \left (d^4 x^4-12 d^2 x^2+24\right )+b^2 \left (d^6 x^6-30 d^4 x^4+360 d^2 x^2-720\right )\right ) \cos (c+d x)}{d^7} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 746, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.17647, size = 826, normalized size = 3.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57149, size = 333, normalized size = 1.41 \begin{align*} -\frac{{\left (b^{2} d^{6} x^{6} - 2 \, a^{2} d^{4} + 2 \,{\left (a b d^{6} - 15 \, b^{2} d^{4}\right )} x^{4} + 48 \, a b d^{2} +{\left (a^{2} d^{6} - 24 \, a b d^{4} + 360 \, b^{2} d^{2}\right )} x^{2} - 720 \, b^{2}\right )} \cos \left (d x + c\right ) - 2 \,{\left (3 \, b^{2} d^{5} x^{5} + 4 \,{\left (a b d^{5} - 15 \, b^{2} d^{3}\right )} x^{3} +{\left (a^{2} d^{5} - 24 \, a b d^{3} + 360 \, b^{2} d\right )} x\right )} \sin \left (d x + c\right )}{d^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.83652, size = 286, normalized size = 1.21 \begin{align*} \begin{cases} - \frac{a^{2} x^{2} \cos{\left (c + d x \right )}}{d} + \frac{2 a^{2} x \sin{\left (c + d x \right )}}{d^{2}} + \frac{2 a^{2} \cos{\left (c + d x \right )}}{d^{3}} - \frac{2 a b x^{4} \cos{\left (c + d x \right )}}{d} + \frac{8 a b x^{3} \sin{\left (c + d x \right )}}{d^{2}} + \frac{24 a b x^{2} \cos{\left (c + d x \right )}}{d^{3}} - \frac{48 a b x \sin{\left (c + d x \right )}}{d^{4}} - \frac{48 a b \cos{\left (c + d x \right )}}{d^{5}} - \frac{b^{2} x^{6} \cos{\left (c + d x \right )}}{d} + \frac{6 b^{2} x^{5} \sin{\left (c + d x \right )}}{d^{2}} + \frac{30 b^{2} x^{4} \cos{\left (c + d x \right )}}{d^{3}} - \frac{120 b^{2} x^{3} \sin{\left (c + d x \right )}}{d^{4}} - \frac{360 b^{2} x^{2} \cos{\left (c + d x \right )}}{d^{5}} + \frac{720 b^{2} x \sin{\left (c + d x \right )}}{d^{6}} + \frac{720 b^{2} \cos{\left (c + d x \right )}}{d^{7}} & \text{for}\: d \neq 0 \\\left (\frac{a^{2} x^{3}}{3} + \frac{2 a b x^{5}}{5} + \frac{b^{2} x^{7}}{7}\right ) \sin{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1305, size = 219, normalized size = 0.93 \begin{align*} -\frac{{\left (b^{2} d^{6} x^{6} + 2 \, a b d^{6} x^{4} + a^{2} d^{6} x^{2} - 30 \, b^{2} d^{4} x^{4} - 24 \, a b d^{4} x^{2} - 2 \, a^{2} d^{4} + 360 \, b^{2} d^{2} x^{2} + 48 \, a b d^{2} - 720 \, b^{2}\right )} \cos \left (d x + c\right )}{d^{7}} + \frac{2 \,{\left (3 \, b^{2} d^{5} x^{5} + 4 \, a b d^{5} x^{3} + a^{2} d^{5} x - 60 \, b^{2} d^{3} x^{3} - 24 \, a b d^{3} x + 360 \, b^{2} d x\right )} \sin \left (d x + c\right )}{d^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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