Integrand size = 23, antiderivative size = 266 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^{5/2}} \, dx=-\frac {4 b \sqrt {1+(c+d x)^2}}{3 d e^2 \sqrt {e (c+d x)}}+\frac {4 b \sqrt {e (c+d x)} \sqrt {1+(c+d x)^2}}{3 d e^3 (1+c+d x)}-\frac {2 (a+b \text {arcsinh}(c+d x))}{3 d e (e (c+d x))^{3/2}}-\frac {4 b (1+c+d x) \sqrt {\frac {1+(c+d x)^2}{(1+c+d x)^2}} E\left (2 \arctan \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )|\frac {1}{2}\right )}{3 d e^{5/2} \sqrt {1+(c+d x)^2}}+\frac {2 b (1+c+d x) \sqrt {\frac {1+(c+d x)^2}{(1+c+d x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),\frac {1}{2}\right )}{3 d e^{5/2} \sqrt {1+(c+d x)^2}} \]
-2/3*(a+b*arcsinh(d*x+c))/d/e/(e*(d*x+c))^(3/2)-4/3*b*(1+(d*x+c)^2)^(1/2)/ d/e^2/(e*(d*x+c))^(1/2)+4/3*b*(e*(d*x+c))^(1/2)*(1+(d*x+c)^2)^(1/2)/d/e^3/ (d*x+c+1)-4/3*b*(d*x+c+1)*(cos(2*arctan((e*(d*x+c))^(1/2)/e^(1/2)))^2)^(1/ 2)/cos(2*arctan((e*(d*x+c))^(1/2)/e^(1/2)))*EllipticE(sin(2*arctan((e*(d*x +c))^(1/2)/e^(1/2))),1/2*2^(1/2))*((1+(d*x+c)^2)/(d*x+c+1)^2)^(1/2)/d/e^(5 /2)/(1+(d*x+c)^2)^(1/2)+2/3*b*(d*x+c+1)*(cos(2*arctan((e*(d*x+c))^(1/2)/e^ (1/2)))^2)^(1/2)/cos(2*arctan((e*(d*x+c))^(1/2)/e^(1/2)))*EllipticF(sin(2* arctan((e*(d*x+c))^(1/2)/e^(1/2))),1/2*2^(1/2))*((1+(d*x+c)^2)/(d*x+c+1)^2 )^(1/2)/d/e^(5/2)/(1+(d*x+c)^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.22 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^{5/2}} \, dx=-\frac {2 \left (a+b \text {arcsinh}(c+d x)+2 b (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-(c+d x)^2\right )\right )}{3 d e (e (c+d x))^{3/2}} \]
(-2*(a + b*ArcSinh[c + d*x] + 2*b*(c + d*x)*Hypergeometric2F1[-1/4, 1/2, 3 /4, -(c + d*x)^2]))/(3*d*e*(e*(c + d*x))^(3/2))
Time = 0.44 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6274, 6191, 264, 266, 834, 27, 761, 1510}
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.
| \(\displaystyle \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int \frac {a+b \text {arcsinh}(c+d x)}{(e (c+d x))^{5/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {\frac {2 b \int \frac {1}{(e (c+d x))^{3/2} \sqrt {(c+d x)^2+1}}d(c+d x)}{3 e}-\frac {2 (a+b \text {arcsinh}(c+d x))}{3 e (e (c+d x))^{3/2}}}{d}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {\frac {2 b \left (\frac {\int \frac {\sqrt {e (c+d x)}}{\sqrt {(c+d x)^2+1}}d(c+d x)}{e^2}-\frac {2 \sqrt {(c+d x)^2+1}}{e \sqrt {e (c+d x)}}\right )}{3 e}-\frac {2 (a+b \text {arcsinh}(c+d x))}{3 e (e (c+d x))^{3/2}}}{d}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {2 b \left (\frac {2 \int \frac {e (c+d x)}{\sqrt {(c+d x)^2+1}}d\sqrt {e (c+d x)}}{e^3}-\frac {2 \sqrt {(c+d x)^2+1}}{e \sqrt {e (c+d x)}}\right )}{3 e}-\frac {2 (a+b \text {arcsinh}(c+d x))}{3 e (e (c+d x))^{3/2}}}{d}\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle \frac {\frac {2 b \left (\frac {2 \left (e \int \frac {1}{\sqrt {(c+d x)^2+1}}d\sqrt {e (c+d x)}-e \int \frac {e-e (c+d x)}{e \sqrt {(c+d x)^2+1}}d\sqrt {e (c+d x)}\right )}{e^3}-\frac {2 \sqrt {(c+d x)^2+1}}{e \sqrt {e (c+d x)}}\right )}{3 e}-\frac {2 (a+b \text {arcsinh}(c+d x))}{3 e (e (c+d x))^{3/2}}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 b \left (\frac {2 \left (e \int \frac {1}{\sqrt {(c+d x)^2+1}}d\sqrt {e (c+d x)}-\int \frac {e-e (c+d x)}{\sqrt {(c+d x)^2+1}}d\sqrt {e (c+d x)}\right )}{e^3}-\frac {2 \sqrt {(c+d x)^2+1}}{e \sqrt {e (c+d x)}}\right )}{3 e}-\frac {2 (a+b \text {arcsinh}(c+d x))}{3 e (e (c+d x))^{3/2}}}{d}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\frac {2 b \left (\frac {2 \left (\frac {\sqrt {e} (e (c+d x)+e) \sqrt {\frac {e^2 (c+d x)^2+e^2}{(e (c+d x)+e)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),\frac {1}{2}\right )}{2 \sqrt {(c+d x)^2+1}}-\int \frac {e-e (c+d x)}{\sqrt {(c+d x)^2+1}}d\sqrt {e (c+d x)}\right )}{e^3}-\frac {2 \sqrt {(c+d x)^2+1}}{e \sqrt {e (c+d x)}}\right )}{3 e}-\frac {2 (a+b \text {arcsinh}(c+d x))}{3 e (e (c+d x))^{3/2}}}{d}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {\frac {2 b \left (\frac {2 \left (\frac {\sqrt {e} (e (c+d x)+e) \sqrt {\frac {e^2 (c+d x)^2+e^2}{(e (c+d x)+e)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),\frac {1}{2}\right )}{2 \sqrt {(c+d x)^2+1}}-\frac {\sqrt {e} (e (c+d x)+e) \sqrt {\frac {e^2 (c+d x)^2+e^2}{(e (c+d x)+e)^2}} E\left (2 \arctan \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )|\frac {1}{2}\right )}{\sqrt {(c+d x)^2+1}}+\frac {e^2 \sqrt {(c+d x)^2+1} \sqrt {e (c+d x)}}{e (c+d x)+e}\right )}{e^3}-\frac {2 \sqrt {(c+d x)^2+1}}{e \sqrt {e (c+d x)}}\right )}{3 e}-\frac {2 (a+b \text {arcsinh}(c+d x))}{3 e (e (c+d x))^{3/2}}}{d}\) |
((-2*(a + b*ArcSinh[c + d*x]))/(3*e*(e*(c + d*x))^(3/2)) + (2*b*((-2*Sqrt[ 1 + (c + d*x)^2])/(e*Sqrt[e*(c + d*x)]) + (2*((e^2*Sqrt[e*(c + d*x)]*Sqrt[ 1 + (c + d*x)^2])/(e + e*(c + d*x)) - (Sqrt[e]*(e + e*(c + d*x))*Sqrt[(e^2 + e^2*(c + d*x)^2)/(e + e*(c + d*x))^2]*EllipticE[2*ArcTan[Sqrt[e*(c + d* x)]/Sqrt[e]], 1/2])/Sqrt[1 + (c + d*x)^2] + (Sqrt[e]*(e + e*(c + d*x))*Sqr t[(e^2 + e^2*(c + d*x)^2)/(e + e*(c + d*x))^2]*EllipticF[2*ArcTan[Sqrt[e*( c + d*x)]/Sqrt[e]], 1/2])/(2*Sqrt[1 + (c + d*x)^2])))/e^3))/(3*e))/d
3.3.34.3.1 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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S imp[1/q Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Result contains complex when optimal does not.
Time = 0.51 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.76
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 a}{3 \left (d e x +c e \right )^{\frac {3}{2}}}+2 b \left (-\frac {\operatorname {arcsinh}\left (\frac {d e x +c e}{e}\right )}{3 \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {-\frac {2 \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{3 \sqrt {d e x +c e}}+\frac {2 i \sqrt {1-\frac {i \left (d e x +c e \right )}{e}}\, \sqrt {1+\frac {i \left (d e x +c e \right )}{e}}\, \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )\right )}{3 e \sqrt {\frac {i}{e}}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{d e}\) | \(202\) |
| default | \(\frac {-\frac {2 a}{3 \left (d e x +c e \right )^{\frac {3}{2}}}+2 b \left (-\frac {\operatorname {arcsinh}\left (\frac {d e x +c e}{e}\right )}{3 \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {-\frac {2 \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{3 \sqrt {d e x +c e}}+\frac {2 i \sqrt {1-\frac {i \left (d e x +c e \right )}{e}}\, \sqrt {1+\frac {i \left (d e x +c e \right )}{e}}\, \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )\right )}{3 e \sqrt {\frac {i}{e}}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{d e}\) | \(202\) |
| parts | \(-\frac {2 a}{3 \left (d e x +c e \right )^{\frac {3}{2}} d e}+\frac {2 b \left (-\frac {\operatorname {arcsinh}\left (\frac {d e x +c e}{e}\right )}{3 \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {-\frac {2 \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{3 \sqrt {d e x +c e}}+\frac {2 i \sqrt {1-\frac {i \left (d e x +c e \right )}{e}}\, \sqrt {1+\frac {i \left (d e x +c e \right )}{e}}\, \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )\right )}{3 e \sqrt {\frac {i}{e}}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{d e}\) | \(207\) |
2/d/e*(-1/3*a/(d*e*x+c*e)^(3/2)+b*(-1/3/(d*e*x+c*e)^(3/2)*arcsinh(1/e*(d*e *x+c*e))+2/3/e*(-(1/e^2*(d*e*x+c*e)^2+1)^(1/2)/(d*e*x+c*e)^(1/2)+I/e/(I/e) ^(1/2)*(1-I/e*(d*e*x+c*e))^(1/2)*(1+I/e*(d*e*x+c*e))^(1/2)/(1/e^2*(d*e*x+c *e)^2+1)^(1/2)*(EllipticF((d*e*x+c*e)^(1/2)*(I/e)^(1/2),I)-EllipticE((d*e* x+c*e)^(1/2)*(I/e)^(1/2),I)))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.68 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^{5/2}} \, dx=-\frac {2 \, {\left (\sqrt {d e x + c e} b d \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) + \sqrt {d e x + c e} a d + 2 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \sqrt {d^{3} e} {\rm weierstrassZeta}\left (-\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (-\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right ) + 2 \, {\left (b d^{2} x + b c d\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} \sqrt {d e x + c e}\right )}}{3 \, {\left (d^{4} e^{3} x^{2} + 2 \, c d^{3} e^{3} x + c^{2} d^{2} e^{3}\right )}} \]
-2/3*(sqrt(d*e*x + c*e)*b*d*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1 )) + sqrt(d*e*x + c*e)*a*d + 2*(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*sqrt(d^3*e) *weierstrassZeta(-4/d^2, 0, weierstrassPInverse(-4/d^2, 0, (d*x + c)/d)) + 2*(b*d^2*x + b*c*d)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*sqrt(d*e*x + c*e))/ (d^4*e^3*x^2 + 2*c*d^3*e^3*x + c^2*d^2*e^3)
\[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^{5/2}} \, dx=\int \frac {a + b \operatorname {asinh}{\left (c + d x \right )}}{\left (e \left (c + d x\right )\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^{5/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^{5/2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^{5/2}} \,d x \]