Integrand size = 23, antiderivative size = 177 \[ \int (c e+d e x)^{5/2} (a+b \text {arcsinh}(c+d x)) \, dx=\frac {20 b e^2 \sqrt {e (c+d x)} \sqrt {1+(c+d x)^2}}{147 d}-\frac {4 b (e (c+d x))^{5/2} \sqrt {1+(c+d x)^2}}{49 d}+\frac {2 (e (c+d x))^{7/2} (a+b \text {arcsinh}(c+d x))}{7 d e}-\frac {10 b e^{5/2} (1+c+d x) \sqrt {\frac {1+(c+d x)^2}{(1+c+d x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),\frac {1}{2}\right )}{147 d \sqrt {1+(c+d x)^2}} \]
2/7*(e*(d*x+c))^(7/2)*(a+b*arcsinh(d*x+c))/d/e-4/49*b*(e*(d*x+c))^(5/2)*(1 +(d*x+c)^2)^(1/2)/d+20/147*b*e^2*(e*(d*x+c))^(1/2)*(1+(d*x+c)^2)^(1/2)/d-1 0/147*b*e^(5/2)*(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/(1+( d*x+c)^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.13 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.64 \[ \int (c e+d e x)^{5/2} (a+b \text {arcsinh}(c+d x)) \, dx=\frac {2 (e (c+d x))^{5/2} \left (21 a (c+d x)^3+10 b \sqrt {1+(c+d x)^2}-6 b (c+d x)^2 \sqrt {1+(c+d x)^2}+21 b (c+d x)^3 \text {arcsinh}(c+d x)-10 b \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-(c+d x)^2\right )\right )}{147 d (c+d x)^2} \]
(2*(e*(c + d*x))^(5/2)*(21*a*(c + d*x)^3 + 10*b*Sqrt[1 + (c + d*x)^2] - 6* b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2] + 21*b*(c + d*x)^3*ArcSinh[c + d*x] - 10*b*Hypergeometric2F1[1/4, 1/2, 5/4, -(c + d*x)^2]))/(147*d*(c + d*x)^2)
Time = 0.35 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6274, 6191, 262, 262, 266, 761}
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.
\(\displaystyle \int (c e+d e x)^{5/2} (a+b \text {arcsinh}(c+d x)) \, dx\) |
\(\Big \downarrow \) 6274 |
\(\displaystyle \frac {\int (e (c+d x))^{5/2} (a+b \text {arcsinh}(c+d x))d(c+d x)}{d}\) |
\(\Big \downarrow \) 6191 |
\(\displaystyle \frac {\frac {2 (e (c+d x))^{7/2} (a+b \text {arcsinh}(c+d x))}{7 e}-\frac {2 b \int \frac {(e (c+d x))^{7/2}}{\sqrt {(c+d x)^2+1}}d(c+d x)}{7 e}}{d}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {\frac {2 (e (c+d x))^{7/2} (a+b \text {arcsinh}(c+d x))}{7 e}-\frac {2 b \left (\frac {2}{7} e \sqrt {(c+d x)^2+1} (e (c+d x))^{5/2}-\frac {5}{7} e^2 \int \frac {(e (c+d x))^{3/2}}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )}{7 e}}{d}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {\frac {2 (e (c+d x))^{7/2} (a+b \text {arcsinh}(c+d x))}{7 e}-\frac {2 b \left (\frac {2}{7} e \sqrt {(c+d x)^2+1} (e (c+d x))^{5/2}-\frac {5}{7} e^2 \left (\frac {2}{3} e \sqrt {(c+d x)^2+1} \sqrt {e (c+d x)}-\frac {1}{3} e^2 \int \frac {1}{\sqrt {e (c+d x)} \sqrt {(c+d x)^2+1}}d(c+d x)\right )\right )}{7 e}}{d}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {\frac {2 (e (c+d x))^{7/2} (a+b \text {arcsinh}(c+d x))}{7 e}-\frac {2 b \left (\frac {2}{7} e \sqrt {(c+d x)^2+1} (e (c+d x))^{5/2}-\frac {5}{7} e^2 \left (\frac {2}{3} e \sqrt {(c+d x)^2+1} \sqrt {e (c+d x)}-\frac {2}{3} e \int \frac {1}{\sqrt {(c+d x)^2+1}}d\sqrt {e (c+d x)}\right )\right )}{7 e}}{d}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {\frac {2 (e (c+d x))^{7/2} (a+b \text {arcsinh}(c+d x))}{7 e}-\frac {2 b \left (\frac {2}{7} e \sqrt {(c+d x)^2+1} (e (c+d x))^{5/2}-\frac {5}{7} e^2 \left (\frac {2}{3} e \sqrt {(c+d x)^2+1} \sqrt {e (c+d x)}-\frac {\sqrt {e} (e (c+d x)+e) \sqrt {\frac {e^2 (c+d x)^2+e^2}{(e (c+d x)+e)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),\frac {1}{2}\right )}{3 \sqrt {(c+d x)^2+1}}\right )\right )}{7 e}}{d}\) |
((2*(e*(c + d*x))^(7/2)*(a + b*ArcSinh[c + d*x]))/(7*e) - (2*b*((2*e*(e*(c + d*x))^(5/2)*Sqrt[1 + (c + d*x)^2])/7 - (5*e^2*((2*e*Sqrt[e*(c + d*x)]*S qrt[1 + (c + d*x)^2])/3 - (Sqrt[e]*(e + e*(c + d*x))*Sqrt[(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])/(3*Sqrt[1 + (c + d*x)^2])))/7))/(7*e))/d
3.3.29.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && 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[((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 = 1.29 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.20
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (d e x +c e \right )^{\frac {7}{2}} a}{7}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {7}{2}} \operatorname {arcsinh}\left (\frac {d e x +c e}{e}\right )}{7}-\frac {2 \left (\frac {e^{2} \left (d e x +c e \right )^{\frac {5}{2}} \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{7}-\frac {5 e^{4} \sqrt {d e x +c e}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{21}+\frac {5 e^{4} \sqrt {1-\frac {i \left (d e x +c e \right )}{e}}\, \sqrt {1+\frac {i \left (d e x +c e \right )}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )}{21 \sqrt {\frac {i}{e}}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{7 e}\right )}{d e}\) | \(212\) |
default | \(\frac {\frac {2 \left (d e x +c e \right )^{\frac {7}{2}} a}{7}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {7}{2}} \operatorname {arcsinh}\left (\frac {d e x +c e}{e}\right )}{7}-\frac {2 \left (\frac {e^{2} \left (d e x +c e \right )^{\frac {5}{2}} \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{7}-\frac {5 e^{4} \sqrt {d e x +c e}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{21}+\frac {5 e^{4} \sqrt {1-\frac {i \left (d e x +c e \right )}{e}}\, \sqrt {1+\frac {i \left (d e x +c e \right )}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )}{21 \sqrt {\frac {i}{e}}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{7 e}\right )}{d e}\) | \(212\) |
parts | \(\frac {2 a \left (d e x +c e \right )^{\frac {7}{2}}}{7 d e}+\frac {2 b \left (\frac {\left (d e x +c e \right )^{\frac {7}{2}} \operatorname {arcsinh}\left (\frac {d e x +c e}{e}\right )}{7}-\frac {2 \left (\frac {e^{2} \left (d e x +c e \right )^{\frac {5}{2}} \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{7}-\frac {5 e^{4} \sqrt {d e x +c e}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{21}+\frac {5 e^{4} \sqrt {1-\frac {i \left (d e x +c e \right )}{e}}\, \sqrt {1+\frac {i \left (d e x +c e \right )}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )}{21 \sqrt {\frac {i}{e}}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{7 e}\right )}{d e}\) | \(217\) |
2/d/e*(1/7*(d*e*x+c*e)^(7/2)*a+b*(1/7*(d*e*x+c*e)^(7/2)*arcsinh(1/e*(d*e*x +c*e))-2/7/e*(1/7*e^2*(d*e*x+c*e)^(5/2)*(1/e^2*(d*e*x+c*e)^2+1)^(1/2)-5/21 *e^4*(d*e*x+c*e)^(1/2)*(1/e^2*(d*e*x+c*e)^2+1)^(1/2)+5/21*e^4/(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))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.44 \[ \int (c e+d e x)^{5/2} (a+b \text {arcsinh}(c+d x)) \, dx=-\frac {2 \, {\left (10 \, \sqrt {d^{3} e} b e^{2} {\rm weierstrassPInverse}\left (-\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right ) - 21 \, {\left (b d^{5} e^{2} x^{3} + 3 \, b c d^{4} e^{2} x^{2} + 3 \, b c^{2} d^{3} e^{2} x + b c^{3} d^{2} e^{2}\right )} \sqrt {d e x + c e} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) + 2 \, {\left (3 \, b d^{4} e^{2} x^{2} + 6 \, b c d^{3} e^{2} x + {\left (3 \, b c^{2} - 5 \, b\right )} d^{2} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} \sqrt {d e x + c e} - 21 \, {\left (a d^{5} e^{2} x^{3} + 3 \, a c d^{4} e^{2} x^{2} + 3 \, a c^{2} d^{3} e^{2} x + a c^{3} d^{2} e^{2}\right )} \sqrt {d e x + c e}\right )}}{147 \, d^{3}} \]
-2/147*(10*sqrt(d^3*e)*b*e^2*weierstrassPInverse(-4/d^2, 0, (d*x + c)/d) - 21*(b*d^5*e^2*x^3 + 3*b*c*d^4*e^2*x^2 + 3*b*c^2*d^3*e^2*x + b*c^3*d^2*e^2 )*sqrt(d*e*x + c*e)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) + 2*( 3*b*d^4*e^2*x^2 + 6*b*c*d^3*e^2*x + (3*b*c^2 - 5*b)*d^2*e^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*sqrt(d*e*x + c*e) - 21*(a*d^5*e^2*x^3 + 3*a*c*d^4*e^2 *x^2 + 3*a*c^2*d^3*e^2*x + a*c^3*d^2*e^2)*sqrt(d*e*x + c*e))/d^3
\[ \int (c e+d e x)^{5/2} (a+b \text {arcsinh}(c+d x)) \, dx=\int \left (e \left (c + d x\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )\, dx \]
Exception generated. \[ \int (c e+d e x)^{5/2} (a+b \text {arcsinh}(c+d x)) \, dx=\text {Exception raised: ValueError} \]
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
\[ \int (c e+d e x)^{5/2} (a+b \text {arcsinh}(c+d x)) \, dx=\int { {\left (d e x + c e\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )} \,d x } \]
Timed out. \[ \int (c e+d e x)^{5/2} (a+b \text {arcsinh}(c+d x)) \, dx=\int {\left (c\,e+d\,e\,x\right )}^{5/2}\,\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right ) \,d x \]