Integrand size = 20, antiderivative size = 227 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+e x^2\right )^{7/2}} \, dx=-\frac {b c \sqrt {1+c^2 x^2}}{15 d \left (c^2 d-e\right ) \left (d+e x^2\right )^{3/2}}-\frac {2 b c \left (3 c^2 d-2 e\right ) \sqrt {1+c^2 x^2}}{15 d^2 \left (c^2 d-e\right )^2 \sqrt {d+e x^2}}+\frac {x (a+b \text {arcsinh}(c x))}{5 d \left (d+e x^2\right )^{5/2}}+\frac {4 x (a+b \text {arcsinh}(c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {8 x (a+b \text {arcsinh}(c x))}{15 d^3 \sqrt {d+e x^2}}-\frac {8 b \text {arctanh}\left (\frac {\sqrt {e} \sqrt {1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{15 d^3 \sqrt {e}} \]
1/5*x*(a+b*arcsinh(c*x))/d/(e*x^2+d)^(5/2)+4/15*x*(a+b*arcsinh(c*x))/d^2/( e*x^2+d)^(3/2)-8/15*b*arctanh(e^(1/2)*(c^2*x^2+1)^(1/2)/c/(e*x^2+d)^(1/2)) /d^3/e^(1/2)-1/15*b*c*(c^2*x^2+1)^(1/2)/d/(c^2*d-e)/(e*x^2+d)^(3/2)+8/15*x *(a+b*arcsinh(c*x))/d^3/(e*x^2+d)^(1/2)-2/15*b*c*(3*c^2*d-2*e)*(c^2*x^2+1) ^(1/2)/d^2/(c^2*d-e)^2/(e*x^2+d)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.29 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.84 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+e x^2\right )^{7/2}} \, dx=\frac {a x \left (15 d^2+20 d e x^2+8 e^2 x^4\right )-\frac {b c d \sqrt {1+c^2 x^2} \left (d+e x^2\right ) \left (-e \left (5 d+4 e x^2\right )+c^2 d \left (7 d+6 e x^2\right )\right )}{\left (-c^2 d+e\right )^2}-4 b c x^2 \left (d+e x^2\right )^2 \sqrt {1+\frac {e x^2}{d}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,-c^2 x^2,-\frac {e x^2}{d}\right )+b x \left (15 d^2+20 d e x^2+8 e^2 x^4\right ) \text {arcsinh}(c x)}{15 d^3 \left (d+e x^2\right )^{5/2}} \]
(a*x*(15*d^2 + 20*d*e*x^2 + 8*e^2*x^4) - (b*c*d*Sqrt[1 + c^2*x^2]*(d + e*x ^2)*(-(e*(5*d + 4*e*x^2)) + c^2*d*(7*d + 6*e*x^2)))/(-(c^2*d) + e)^2 - 4*b *c*x^2*(d + e*x^2)^2*Sqrt[1 + (e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2, -(c^2*x^ 2), -((e*x^2)/d)] + b*x*(15*d^2 + 20*d*e*x^2 + 8*e^2*x^4)*ArcSinh[c*x])/(1 5*d^3*(d + e*x^2)^(5/2))
Time = 1.02 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6207, 27, 7266, 1193, 27, 87, 66, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{\left (d+e x^2\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 6207 |
| \(\displaystyle -b c \int \frac {x \left (8 e^2 x^4+20 d e x^2+15 d^2\right )}{15 d^3 \sqrt {c^2 x^2+1} \left (e x^2+d\right )^{5/2}}dx+\frac {8 x (a+b \text {arcsinh}(c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \text {arcsinh}(c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \text {arcsinh}(c x))}{5 d \left (d+e x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c \int \frac {x \left (8 e^2 x^4+20 d e x^2+15 d^2\right )}{\sqrt {c^2 x^2+1} \left (e x^2+d\right )^{5/2}}dx}{15 d^3}+\frac {8 x (a+b \text {arcsinh}(c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \text {arcsinh}(c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \text {arcsinh}(c x))}{5 d \left (d+e x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 7266 |
| \(\displaystyle -\frac {b c \int \frac {8 e^2 x^4+20 d e x^2+15 d^2}{\sqrt {c^2 x^2+1} \left (e x^2+d\right )^{5/2}}dx^2}{30 d^3}+\frac {8 x (a+b \text {arcsinh}(c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \text {arcsinh}(c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \text {arcsinh}(c x))}{5 d \left (d+e x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 1193 |
| \(\displaystyle -\frac {b c \left (\frac {2 \int \frac {3 \left (4 \left (c^2 d-e\right ) e x^2+d \left (7 c^2 d-6 e\right )\right )}{\sqrt {c^2 x^2+1} \left (e x^2+d\right )^{3/2}}dx^2}{3 \left (c^2 d-e\right )}+\frac {2 d^2 \sqrt {c^2 x^2+1}}{\left (c^2 d-e\right ) \left (d+e x^2\right )^{3/2}}\right )}{30 d^3}+\frac {8 x (a+b \text {arcsinh}(c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \text {arcsinh}(c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \text {arcsinh}(c x))}{5 d \left (d+e x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c \left (\frac {2 \int \frac {4 \left (c^2 d-e\right ) e x^2+d \left (7 c^2 d-6 e\right )}{\sqrt {c^2 x^2+1} \left (e x^2+d\right )^{3/2}}dx^2}{c^2 d-e}+\frac {2 d^2 \sqrt {c^2 x^2+1}}{\left (c^2 d-e\right ) \left (d+e x^2\right )^{3/2}}\right )}{30 d^3}+\frac {8 x (a+b \text {arcsinh}(c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \text {arcsinh}(c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \text {arcsinh}(c x))}{5 d \left (d+e x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {b c \left (\frac {2 \left (4 \left (c^2 d-e\right ) \int \frac {1}{\sqrt {c^2 x^2+1} \sqrt {e x^2+d}}dx^2+\frac {2 d \sqrt {c^2 x^2+1} \left (3 c^2 d-2 e\right )}{\left (c^2 d-e\right ) \sqrt {d+e x^2}}\right )}{c^2 d-e}+\frac {2 d^2 \sqrt {c^2 x^2+1}}{\left (c^2 d-e\right ) \left (d+e x^2\right )^{3/2}}\right )}{30 d^3}+\frac {8 x (a+b \text {arcsinh}(c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \text {arcsinh}(c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \text {arcsinh}(c x))}{5 d \left (d+e x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle -\frac {b c \left (\frac {2 \left (8 \left (c^2 d-e\right ) \int \frac {1}{c^2-e x^4}d\frac {\sqrt {c^2 x^2+1}}{\sqrt {e x^2+d}}+\frac {2 d \sqrt {c^2 x^2+1} \left (3 c^2 d-2 e\right )}{\left (c^2 d-e\right ) \sqrt {d+e x^2}}\right )}{c^2 d-e}+\frac {2 d^2 \sqrt {c^2 x^2+1}}{\left (c^2 d-e\right ) \left (d+e x^2\right )^{3/2}}\right )}{30 d^3}+\frac {8 x (a+b \text {arcsinh}(c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \text {arcsinh}(c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \text {arcsinh}(c x))}{5 d \left (d+e x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {8 x (a+b \text {arcsinh}(c x))}{15 d^3 \sqrt {d+e x^2}}+\frac {4 x (a+b \text {arcsinh}(c x))}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac {x (a+b \text {arcsinh}(c x))}{5 d \left (d+e x^2\right )^{5/2}}-\frac {b c \left (\frac {2 \left (\frac {8 \left (c^2 d-e\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {c^2 x^2+1}}{c \sqrt {d+e x^2}}\right )}{c \sqrt {e}}+\frac {2 d \sqrt {c^2 x^2+1} \left (3 c^2 d-2 e\right )}{\left (c^2 d-e\right ) \sqrt {d+e x^2}}\right )}{c^2 d-e}+\frac {2 d^2 \sqrt {c^2 x^2+1}}{\left (c^2 d-e\right ) \left (d+e x^2\right )^{3/2}}\right )}{30 d^3}\) |
(x*(a + b*ArcSinh[c*x]))/(5*d*(d + e*x^2)^(5/2)) + (4*x*(a + b*ArcSinh[c*x ]))/(15*d^2*(d + e*x^2)^(3/2)) + (8*x*(a + b*ArcSinh[c*x]))/(15*d^3*Sqrt[d + e*x^2]) - (b*c*((2*d^2*Sqrt[1 + c^2*x^2])/((c^2*d - e)*(d + e*x^2)^(3/2 )) + (2*((2*d*(3*c^2*d - 2*e)*Sqrt[1 + c^2*x^2])/((c^2*d - e)*Sqrt[d + e*x ^2]) + (8*(c^2*d - e)*ArcTanh[(Sqrt[e]*Sqrt[1 + c^2*x^2])/(c*Sqrt[d + e*x^ 2])])/(c*Sqrt[e])))/(c^2*d - e)))/(30*d^3)
3.7.51.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + b*x + c*x^2)^p, d + e*x, x]}, Simp[R*(d + e*x)^(m + 1)*((f + g*x)^(n + 1)/((m + 1)*(e*f - d*g)) ), x] + Simp[1/((m + 1)*(e*f - d*g)) Int[(d + e*x)^(m + 1)*(f + g*x)^n*Ex pandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x], x]] /; FreeQ[{a , b, c, d, e, f, g, n}, x] && IGtQ[p, 0] && ILtQ[2*m, -2] && !IntegerQ[n] && !(EqQ[m, -2] && EqQ[p, 1] && EqQ[2*c*d - b*e, 0])
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symb ol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcSinh[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[e, c^2*d] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])
Int[(u_)*(x_)^(m_.), x_Symbol] :> Simp[1/(m + 1) Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /; FreeQ[m, x] && NeQ[m, -1] && Function OfQ[x^(m + 1), u, x]
\[\int \frac {a +b \,\operatorname {arcsinh}\left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {7}{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 670 vs. \(2 (193) = 386\).
Time = 0.39 (sec) , antiderivative size = 1354, normalized size of antiderivative = 5.96 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+e x^2\right )^{7/2}} \, dx=\text {Too large to display} \]
[1/15*(2*(b*c^4*d^5 - 2*b*c^2*d^4*e + (b*c^4*d^2*e^3 - 2*b*c^2*d*e^4 + b*e ^5)*x^6 + b*d^3*e^2 + 3*(b*c^4*d^3*e^2 - 2*b*c^2*d^2*e^3 + b*d*e^4)*x^4 + 3*(b*c^4*d^4*e - 2*b*c^2*d^3*e^2 + b*d^2*e^3)*x^2)*sqrt(e)*log(8*c^4*e^2*x ^4 + c^4*d^2 + 6*c^2*d*e + 8*(c^4*d*e + c^2*e^2)*x^2 - 4*(2*c^3*e*x^2 + c^ 3*d + c*e)*sqrt(c^2*x^2 + 1)*sqrt(e*x^2 + d)*sqrt(e) + e^2) + (8*(b*c^4*d^ 2*e^3 - 2*b*c^2*d*e^4 + b*e^5)*x^5 + 20*(b*c^4*d^3*e^2 - 2*b*c^2*d^2*e^3 + b*d*e^4)*x^3 + 15*(b*c^4*d^4*e - 2*b*c^2*d^3*e^2 + b*d^2*e^3)*x)*sqrt(e*x ^2 + d)*log(c*x + sqrt(c^2*x^2 + 1)) + (8*(a*c^4*d^2*e^3 - 2*a*c^2*d*e^4 + a*e^5)*x^5 + 20*(a*c^4*d^3*e^2 - 2*a*c^2*d^2*e^3 + a*d*e^4)*x^3 + 15*(a*c ^4*d^4*e - 2*a*c^2*d^3*e^2 + a*d^2*e^3)*x - (7*b*c^3*d^4*e - 5*b*c*d^3*e^2 + 2*(3*b*c^3*d^2*e^3 - 2*b*c*d*e^4)*x^4 + (13*b*c^3*d^3*e^2 - 9*b*c*d^2*e ^3)*x^2)*sqrt(c^2*x^2 + 1))*sqrt(e*x^2 + d))/(c^4*d^8*e - 2*c^2*d^7*e^2 + d^6*e^3 + (c^4*d^5*e^4 - 2*c^2*d^4*e^5 + d^3*e^6)*x^6 + 3*(c^4*d^6*e^3 - 2 *c^2*d^5*e^4 + d^4*e^5)*x^4 + 3*(c^4*d^7*e^2 - 2*c^2*d^6*e^3 + d^5*e^4)*x^ 2), 1/15*(4*(b*c^4*d^5 - 2*b*c^2*d^4*e + (b*c^4*d^2*e^3 - 2*b*c^2*d*e^4 + b*e^5)*x^6 + b*d^3*e^2 + 3*(b*c^4*d^3*e^2 - 2*b*c^2*d^2*e^3 + b*d*e^4)*x^4 + 3*(b*c^4*d^4*e - 2*b*c^2*d^3*e^2 + b*d^2*e^3)*x^2)*sqrt(-e)*arctan(1/2* (2*c^2*e*x^2 + c^2*d + e)*sqrt(c^2*x^2 + 1)*sqrt(e*x^2 + d)*sqrt(-e)/(c^3* e^2*x^4 + c*d*e + (c^3*d*e + c*e^2)*x^2)) + (8*(b*c^4*d^2*e^3 - 2*b*c^2*d* e^4 + b*e^5)*x^5 + 20*(b*c^4*d^3*e^2 - 2*b*c^2*d^2*e^3 + b*d*e^4)*x^3 +...
Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+e x^2\right )^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \]
1/15*a*(8*x/(sqrt(e*x^2 + d)*d^3) + 4*x/((e*x^2 + d)^(3/2)*d^2) + 3*x/((e* x^2 + d)^(5/2)*d)) + b*integrate(log(c*x + sqrt(c^2*x^2 + 1))/(e*x^2 + d)^ (7/2), x)
\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+e x^2\right )^{7/2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^{7/2}} \,d x \]