Optimal. Leaf size=346 \[ -\frac{2 c^2 \sinh \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{10 (c+d x)^2 \sinh \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}+\frac{12 c (c+d x) \sinh \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{120 (c+d x) \sinh \left (a+b \sqrt{c+d x}\right )}{b^4 d^3}+\frac{24 c \sinh \left (a+b \sqrt{c+d x}\right )}{b^4 d^3}-\frac{240 \sinh \left (a+b \sqrt{c+d x}\right )}{b^6 d^3}+\frac{40 (c+d x)^{3/2} \cosh \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}-\frac{24 c \sqrt{c+d x} \cosh \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}+\frac{240 \sqrt{c+d x} \cosh \left (a+b \sqrt{c+d x}\right )}{b^5 d^3}+\frac{2 c^2 \sqrt{c+d x} \cosh \left (a+b \sqrt{c+d x}\right )}{b d^3}+\frac{2 (c+d x)^{5/2} \cosh \left (a+b \sqrt{c+d x}\right )}{b d^3}-\frac{4 c (c+d x)^{3/2} \cosh \left (a+b \sqrt{c+d x}\right )}{b d^3} \]
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Rubi [A] time = 0.418116, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5364, 5286, 3296, 2637} \[ -\frac{2 c^2 \sinh \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{10 (c+d x)^2 \sinh \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}+\frac{12 c (c+d x) \sinh \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{120 (c+d x) \sinh \left (a+b \sqrt{c+d x}\right )}{b^4 d^3}+\frac{24 c \sinh \left (a+b \sqrt{c+d x}\right )}{b^4 d^3}-\frac{240 \sinh \left (a+b \sqrt{c+d x}\right )}{b^6 d^3}+\frac{40 (c+d x)^{3/2} \cosh \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}-\frac{24 c \sqrt{c+d x} \cosh \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}+\frac{240 \sqrt{c+d x} \cosh \left (a+b \sqrt{c+d x}\right )}{b^5 d^3}+\frac{2 c^2 \sqrt{c+d x} \cosh \left (a+b \sqrt{c+d x}\right )}{b d^3}+\frac{2 (c+d x)^{5/2} \cosh \left (a+b \sqrt{c+d x}\right )}{b d^3}-\frac{4 c (c+d x)^{3/2} \cosh \left (a+b \sqrt{c+d x}\right )}{b d^3} \]
Antiderivative was successfully verified.
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Rule 5364
Rule 5286
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x^2 \sinh \left (a+b \sqrt{c+d x}\right ) \, dx &=\frac{\operatorname{Subst}\left (\int (-c+x)^2 \sinh \left (a+b \sqrt{x}\right ) \, dx,x,c+d x\right )}{d^3}\\ &=\frac{2 \operatorname{Subst}\left (\int x \left (c-x^2\right )^2 \sinh (a+b x) \, dx,x,\sqrt{c+d x}\right )}{d^3}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (c^2 x \sinh (a+b x)-2 c x^3 \sinh (a+b x)+x^5 \sinh (a+b x)\right ) \, dx,x,\sqrt{c+d x}\right )}{d^3}\\ &=\frac{2 \operatorname{Subst}\left (\int x^5 \sinh (a+b x) \, dx,x,\sqrt{c+d x}\right )}{d^3}-\frac{(4 c) \operatorname{Subst}\left (\int x^3 \sinh (a+b x) \, dx,x,\sqrt{c+d x}\right )}{d^3}+\frac{\left (2 c^2\right ) \operatorname{Subst}\left (\int x \sinh (a+b x) \, dx,x,\sqrt{c+d x}\right )}{d^3}\\ &=\frac{2 c^2 \sqrt{c+d x} \cosh \left (a+b \sqrt{c+d x}\right )}{b d^3}-\frac{4 c (c+d x)^{3/2} \cosh \left (a+b \sqrt{c+d x}\right )}{b d^3}+\frac{2 (c+d x)^{5/2} \cosh \left (a+b \sqrt{c+d x}\right )}{b d^3}-\frac{10 \operatorname{Subst}\left (\int x^4 \cosh (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b d^3}+\frac{(12 c) \operatorname{Subst}\left (\int x^2 \cosh (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b d^3}-\frac{\left (2 c^2\right ) \operatorname{Subst}\left (\int \cosh (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b d^3}\\ &=\frac{2 c^2 \sqrt{c+d x} \cosh \left (a+b \sqrt{c+d x}\right )}{b d^3}-\frac{4 c (c+d x)^{3/2} \cosh \left (a+b \sqrt{c+d x}\right )}{b d^3}+\frac{2 (c+d x)^{5/2} \cosh \left (a+b \sqrt{c+d x}\right )}{b d^3}-\frac{2 c^2 \sinh \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}+\frac{12 c (c+d x) \sinh \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{10 (c+d x)^2 \sinh \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}+\frac{40 \operatorname{Subst}\left (\int x^3 \sinh (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b^2 d^3}-\frac{(24 c) \operatorname{Subst}\left (\int x \sinh (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b^2 d^3}\\ &=-\frac{24 c \sqrt{c+d x} \cosh \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}+\frac{2 c^2 \sqrt{c+d x} \cosh \left (a+b \sqrt{c+d x}\right )}{b d^3}+\frac{40 (c+d x)^{3/2} \cosh \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}-\frac{4 c (c+d x)^{3/2} \cosh \left (a+b \sqrt{c+d x}\right )}{b d^3}+\frac{2 (c+d x)^{5/2} \cosh \left (a+b \sqrt{c+d x}\right )}{b d^3}-\frac{2 c^2 \sinh \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}+\frac{12 c (c+d x) \sinh \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{10 (c+d x)^2 \sinh \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{120 \operatorname{Subst}\left (\int x^2 \cosh (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b^3 d^3}+\frac{(24 c) \operatorname{Subst}\left (\int \cosh (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b^3 d^3}\\ &=-\frac{24 c \sqrt{c+d x} \cosh \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}+\frac{2 c^2 \sqrt{c+d x} \cosh \left (a+b \sqrt{c+d x}\right )}{b d^3}+\frac{40 (c+d x)^{3/2} \cosh \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}-\frac{4 c (c+d x)^{3/2} \cosh \left (a+b \sqrt{c+d x}\right )}{b d^3}+\frac{2 (c+d x)^{5/2} \cosh \left (a+b \sqrt{c+d x}\right )}{b d^3}+\frac{24 c \sinh \left (a+b \sqrt{c+d x}\right )}{b^4 d^3}-\frac{2 c^2 \sinh \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{120 (c+d x) \sinh \left (a+b \sqrt{c+d x}\right )}{b^4 d^3}+\frac{12 c (c+d x) \sinh \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{10 (c+d x)^2 \sinh \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}+\frac{240 \operatorname{Subst}\left (\int x \sinh (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b^4 d^3}\\ &=\frac{240 \sqrt{c+d x} \cosh \left (a+b \sqrt{c+d x}\right )}{b^5 d^3}-\frac{24 c \sqrt{c+d x} \cosh \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}+\frac{2 c^2 \sqrt{c+d x} \cosh \left (a+b \sqrt{c+d x}\right )}{b d^3}+\frac{40 (c+d x)^{3/2} \cosh \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}-\frac{4 c (c+d x)^{3/2} \cosh \left (a+b \sqrt{c+d x}\right )}{b d^3}+\frac{2 (c+d x)^{5/2} \cosh \left (a+b \sqrt{c+d x}\right )}{b d^3}+\frac{24 c \sinh \left (a+b \sqrt{c+d x}\right )}{b^4 d^3}-\frac{2 c^2 \sinh \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{120 (c+d x) \sinh \left (a+b \sqrt{c+d x}\right )}{b^4 d^3}+\frac{12 c (c+d x) \sinh \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{10 (c+d x)^2 \sinh \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{240 \operatorname{Subst}\left (\int \cosh (a+b x) \, dx,x,\sqrt{c+d x}\right )}{b^5 d^3}\\ &=\frac{240 \sqrt{c+d x} \cosh \left (a+b \sqrt{c+d x}\right )}{b^5 d^3}-\frac{24 c \sqrt{c+d x} \cosh \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}+\frac{2 c^2 \sqrt{c+d x} \cosh \left (a+b \sqrt{c+d x}\right )}{b d^3}+\frac{40 (c+d x)^{3/2} \cosh \left (a+b \sqrt{c+d x}\right )}{b^3 d^3}-\frac{4 c (c+d x)^{3/2} \cosh \left (a+b \sqrt{c+d x}\right )}{b d^3}+\frac{2 (c+d x)^{5/2} \cosh \left (a+b \sqrt{c+d x}\right )}{b d^3}-\frac{240 \sinh \left (a+b \sqrt{c+d x}\right )}{b^6 d^3}+\frac{24 c \sinh \left (a+b \sqrt{c+d x}\right )}{b^4 d^3}-\frac{2 c^2 \sinh \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{120 (c+d x) \sinh \left (a+b \sqrt{c+d x}\right )}{b^4 d^3}+\frac{12 c (c+d x) \sinh \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}-\frac{10 (c+d x)^2 \sinh \left (a+b \sqrt{c+d x}\right )}{b^2 d^3}\\ \end{align*}
Mathematica [A] time = 1.29631, size = 104, normalized size = 0.3 \[ \frac{2 b \sqrt{c+d x} \left (4 b^2 (2 c+5 d x)+b^4 d^2 x^2+120\right ) \cosh \left (a+b \sqrt{c+d x}\right )-2 \left (b^4 d x (4 c+5 d x)+12 b^2 (4 c+5 d x)+120\right ) \sinh \left (a+b \sqrt{c+d x}\right )}{b^6 d^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 831, normalized size = 2.4 \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 [A] time = 1.09458, size = 656, normalized size = 1.9 \begin{align*} \frac{2 \, d^{3} x^{3} \sinh \left (\sqrt{d x + c} b + a\right ) +{\left (\frac{c^{3} e^{\left (\sqrt{d x + c} b + a\right )}}{b} - \frac{c^{3} e^{\left (-\sqrt{d x + c} b - a\right )}}{b} - \frac{3 \,{\left ({\left (d x + c\right )} b^{2} e^{a} - 2 \, \sqrt{d x + c} b e^{a} + 2 \, e^{a}\right )} c^{2} e^{\left (\sqrt{d x + c} b\right )}}{b^{3}} + \frac{3 \,{\left ({\left (d x + c\right )} b^{2} + 2 \, \sqrt{d x + c} b + 2\right )} c^{2} e^{\left (-\sqrt{d x + c} b - a\right )}}{b^{3}} + \frac{3 \,{\left ({\left (d x + c\right )}^{2} b^{4} e^{a} - 4 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{3} e^{a} + 12 \,{\left (d x + c\right )} b^{2} e^{a} - 24 \, \sqrt{d x + c} b e^{a} + 24 \, e^{a}\right )} c e^{\left (\sqrt{d x + c} b\right )}}{b^{5}} - \frac{3 \,{\left ({\left (d x + c\right )}^{2} b^{4} + 4 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{3} + 12 \,{\left (d x + c\right )} b^{2} + 24 \, \sqrt{d x + c} b + 24\right )} c e^{\left (-\sqrt{d x + c} b - a\right )}}{b^{5}} - \frac{{\left ({\left (d x + c\right )}^{3} b^{6} e^{a} - 6 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{5} e^{a} + 30 \,{\left (d x + c\right )}^{2} b^{4} e^{a} - 120 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{3} e^{a} + 360 \,{\left (d x + c\right )} b^{2} e^{a} - 720 \, \sqrt{d x + c} b e^{a} + 720 \, e^{a}\right )} e^{\left (\sqrt{d x + c} b\right )}}{b^{7}} + \frac{{\left ({\left (d x + c\right )}^{3} b^{6} + 6 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{5} + 30 \,{\left (d x + c\right )}^{2} b^{4} + 120 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{3} + 360 \,{\left (d x + c\right )} b^{2} + 720 \, \sqrt{d x + c} b + 720\right )} e^{\left (-\sqrt{d x + c} b - a\right )}}{b^{7}}\right )} b}{6 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09057, size = 251, normalized size = 0.73 \begin{align*} \frac{2 \,{\left ({\left (b^{5} d^{2} x^{2} + 20 \, b^{3} d x + 8 \, b^{3} c + 120 \, b\right )} \sqrt{d x + c} \cosh \left (\sqrt{d x + c} b + a\right ) -{\left (5 \, b^{4} d^{2} x^{2} + 48 \, b^{2} c + 4 \,{\left (b^{4} c + 15 \, b^{2}\right )} d x + 120\right )} \sinh \left (\sqrt{d x + c} b + a\right )\right )}}{b^{6} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.4445, size = 269, normalized size = 0.78 \begin{align*} \begin{cases} \frac{x^{3} \sinh{\left (a \right )}}{3} & \text{for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\\frac{x^{3} \sinh{\left (a + b \sqrt{c} \right )}}{3} & \text{for}\: d = 0 \\\frac{2 x^{2} \sqrt{c + d x} \cosh{\left (a + b \sqrt{c + d x} \right )}}{b d} - \frac{8 c x \sinh{\left (a + b \sqrt{c + d x} \right )}}{b^{2} d^{2}} - \frac{10 x^{2} \sinh{\left (a + b \sqrt{c + d x} \right )}}{b^{2} d} + \frac{16 c \sqrt{c + d x} \cosh{\left (a + b \sqrt{c + d x} \right )}}{b^{3} d^{3}} + \frac{40 x \sqrt{c + d x} \cosh{\left (a + b \sqrt{c + d x} \right )}}{b^{3} d^{2}} - \frac{96 c \sinh{\left (a + b \sqrt{c + d x} \right )}}{b^{4} d^{3}} - \frac{120 x \sinh{\left (a + b \sqrt{c + d x} \right )}}{b^{4} d^{2}} + \frac{240 \sqrt{c + d x} \cosh{\left (a + b \sqrt{c + d x} \right )}}{b^{5} d^{3}} - \frac{240 \sinh{\left (a + b \sqrt{c + d x} \right )}}{b^{6} d^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.90568, size = 2072, normalized size = 5.99 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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