Integrand size = 30, antiderivative size = 984 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\frac {a d \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d+c^2 d x^2}}{3 g \sqrt {1+c^2 x^2}}-\frac {b c d \left (c^2 f^2+g^2\right ) x \sqrt {d+c^2 d x^2}}{g^3 \sqrt {1+c^2 x^2}}+\frac {b c^3 d f x^2 \sqrt {d+c^2 d x^2}}{4 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^3 \sqrt {d+c^2 d x^2}}{9 g \sqrt {1+c^2 x^2}}+\frac {b d \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{g^3}-\frac {c^2 d f x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 g}-\frac {c d f \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d \left (c^2 f^2+g^2\right ) x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g^3 \sqrt {1+c^2 x^2}}-\frac {d \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c g^4 (f+g x) \sqrt {1+c^2 x^2}}+\frac {d \left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c g^2 (f+g x)}-\frac {a d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2} \text {arctanh}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\right )}{g^4 \sqrt {1+c^2 x^2}}+\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^4 \sqrt {1+c^2 x^2}}-\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^4 \sqrt {1+c^2 x^2}}+\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^4 \sqrt {1+c^2 x^2}}-\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^4 \sqrt {1+c^2 x^2}} \]
a*d*(c^2*f^2+g^2)*(c^2*d*x^2+d)^(1/2)/g^3+b*d*(c^2*f^2+g^2)*arcsinh(c*x)*( c^2*d*x^2+d)^(1/2)/g^3-1/2*c^2*d*f*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2 )/g^2+1/3*d*(c^2*x^2+1)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/g-1/3*b*c*d *x*(c^2*d*x^2+d)^(1/2)/g/(c^2*x^2+1)^(1/2)-b*c*d*(c^2*f^2+g^2)*x*(c^2*d*x^ 2+d)^(1/2)/g^3/(c^2*x^2+1)^(1/2)+1/4*b*c^3*d*f*x^2*(c^2*d*x^2+d)^(1/2)/g^2 /(c^2*x^2+1)^(1/2)-1/9*b*c^3*d*x^3*(c^2*d*x^2+d)^(1/2)/g/(c^2*x^2+1)^(1/2) -1/4*c*d*f*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/b/g^2/(c^2*x^2+1)^(1/2 )-1/2*c*d*(c^2*f^2+g^2)*x*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/b/g^3/( c^2*x^2+1)^(1/2)-1/2*d*(c^2*f^2+g^2)^2*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^ (1/2)/b/c/g^4/(g*x+f)/(c^2*x^2+1)^(1/2)-a*d*(c^2*f^2+g^2)^(3/2)*arctanh((- c^2*f*x+g)/(c^2*f^2+g^2)^(1/2)/(c^2*x^2+1)^(1/2))*(c^2*d*x^2+d)^(1/2)/g^4/ (c^2*x^2+1)^(1/2)+b*d*(c^2*f^2+g^2)^(3/2)*arcsinh(c*x)*ln(1+(c*x+(c^2*x^2+ 1)^(1/2))*g/(c*f-(c^2*f^2+g^2)^(1/2)))*(c^2*d*x^2+d)^(1/2)/g^4/(c^2*x^2+1) ^(1/2)-b*d*(c^2*f^2+g^2)^(3/2)*arcsinh(c*x)*ln(1+(c*x+(c^2*x^2+1)^(1/2))*g /(c*f+(c^2*f^2+g^2)^(1/2)))*(c^2*d*x^2+d)^(1/2)/g^4/(c^2*x^2+1)^(1/2)+b*d* (c^2*f^2+g^2)^(3/2)*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2+g^2 )^(1/2)))*(c^2*d*x^2+d)^(1/2)/g^4/(c^2*x^2+1)^(1/2)-b*d*(c^2*f^2+g^2)^(3/2 )*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2+g^2)^(1/2)))*(c^2*d*x ^2+d)^(1/2)/g^4/(c^2*x^2+1)^(1/2)+1/2*d*(c^2*f^2+g^2)*(a+b*arcsinh(c*x))^2 *(c^2*x^2+1)^(1/2)*(c^2*d*x^2+d)^(1/2)/b/c/g^2/(g*x+f)
Result contains complex when optimal does not.
Time = 12.24 (sec) , antiderivative size = 2869, normalized size of antiderivative = 2.92 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\text {Result too large to show} \]
(a*d*Sqrt[d + c^2*d*x^2]*(8*g^2 + c^2*(6*f^2 - 3*f*g*x + 2*g^2*x^2)))/(6*g ^3) + (a*d^(3/2)*(c^2*f^2 + g^2)^(3/2)*Log[f + g*x])/g^4 - (a*c*d^(3/2)*f* (2*c^2*f^2 + 3*g^2)*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]])/(2*g^4) - (a *d^(3/2)*(c^2*f^2 + g^2)^(3/2)*Log[d*(g - c^2*f*x) + Sqrt[d]*Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]])/g^4 + (b*d*Sqrt[d + c^2*d*x^2]*((-2*c*g*x)/Sq rt[1 + c^2*x^2] + 2*g*ArcSinh[c*x] - (c*f*ArcSinh[c*x]^2)/Sqrt[1 + c^2*x^2 ] + (2*((-I)*c*f + g)*(I*c*f + g)*(((-I)*Pi*ArcTanh[(-g + c*f*Tanh[ArcSinh [c*x]/2])/Sqrt[c^2*f^2 + g^2]])/Sqrt[c^2*f^2 + g^2] - (2*ArcCos[((-I)*c*f) /g]*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + (Pi - (2*I)*ArcSinh[c*x])*ArcTanh[((c*f - I*g)*Tan[(Pi + (2*I)* ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + (ArcCos[((-I)*c*f)/g] - (2*I)* ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g ^2]] - (2*I)*ArcTanh[((c*f - I*g)*Tan[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[- (c^2*f^2) - g^2]])*Log[((1/2 - I/2)*Sqrt[-(c^2*f^2) - g^2])/(E^(ArcSinh[c* x]/2)*Sqrt[(-I)*g]*Sqrt[c*(f + g*x)])] + (ArcCos[((-I)*c*f)/g] + (2*I)*(Ar cTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2 ]] + ArcTanh[((c*f - I*g)*Tan[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2 ) - g^2]]))*Log[((1/2 + I/2)*E^(ArcSinh[c*x]/2)*Sqrt[-(c^2*f^2) - g^2])/(S qrt[(-I)*g]*Sqrt[c*(f + g*x)])] - (ArcCos[((-I)*c*f)/g] + (2*I)*ArcTanh[(( c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]])*L...
Time = 2.24 (sec) , antiderivative size = 641, normalized size of antiderivative = 0.65, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6260, 6255, 2009}
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.
\(\displaystyle \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx\) |
\(\Big \downarrow \) 6260 |
\(\displaystyle \frac {d \sqrt {c^2 d x^2+d} \int \frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{f+g x}dx}{\sqrt {c^2 x^2+1}}\) |
\(\Big \downarrow \) 6255 |
\(\displaystyle \frac {d \sqrt {c^2 d x^2+d} \int \left (\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) c^2}{g}-\frac {f \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) c^2}{g^2}+\frac {\left (c^2 f^2+g^2\right ) \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{g^2 (f+g x)}\right )dx}{\sqrt {c^2 x^2+1}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right ) \left (c^2 f^2+g^2\right ) (a+b \text {arcsinh}(c x))^2}{2 b c g^2 (f+g x)}-\frac {\left (c^2 f^2+g^2\right )^2 (a+b \text {arcsinh}(c x))^2}{2 b c g^4 (f+g x)}-\frac {c x \left (c^2 f^2+g^2\right ) (a+b \text {arcsinh}(c x))^2}{2 b g^3}-\frac {c^2 f x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 g^2}+\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 g}-\frac {c f (a+b \text {arcsinh}(c x))^2}{4 b g^2}-\frac {a \left (c^2 f^2+g^2\right )^{3/2} \text {arctanh}\left (\frac {g-c^2 f x}{\sqrt {c^2 x^2+1} \sqrt {c^2 f^2+g^2}}\right )}{g^4}+\frac {a \sqrt {c^2 x^2+1} \left (c^2 f^2+g^2\right )}{g^3}+\frac {b \left (c^2 f^2+g^2\right )^{3/2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^4}-\frac {b \left (c^2 f^2+g^2\right )^{3/2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^4}+\frac {b \text {arcsinh}(c x) \left (c^2 f^2+g^2\right )^{3/2} \log \left (\frac {g e^{\text {arcsinh}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{g^4}-\frac {b \text {arcsinh}(c x) \left (c^2 f^2+g^2\right )^{3/2} \log \left (\frac {g e^{\text {arcsinh}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{g^4}+\frac {b \sqrt {c^2 x^2+1} \text {arcsinh}(c x) \left (c^2 f^2+g^2\right )}{g^3}+\frac {b c^3 f x^2}{4 g^2}-\frac {b c^3 x^3}{9 g}-\frac {b c x \left (c^2 f^2+g^2\right )}{g^3}-\frac {b c x}{3 g}\right )}{\sqrt {c^2 x^2+1}}\) |
(d*Sqrt[d + c^2*d*x^2]*(-1/3*(b*c*x)/g - (b*c*(c^2*f^2 + g^2)*x)/g^3 + (b* c^3*f*x^2)/(4*g^2) - (b*c^3*x^3)/(9*g) + (a*(c^2*f^2 + g^2)*Sqrt[1 + c^2*x ^2])/g^3 + (b*(c^2*f^2 + g^2)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/g^3 - (c^2*f *x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*g^2) + ((1 + c^2*x^2)^(3/2)* (a + b*ArcSinh[c*x]))/(3*g) - (c*f*(a + b*ArcSinh[c*x])^2)/(4*b*g^2) - (c* (c^2*f^2 + g^2)*x*(a + b*ArcSinh[c*x])^2)/(2*b*g^3) - ((c^2*f^2 + g^2)^2*( a + b*ArcSinh[c*x])^2)/(2*b*c*g^4*(f + g*x)) + ((c^2*f^2 + g^2)*(1 + c^2*x ^2)*(a + b*ArcSinh[c*x])^2)/(2*b*c*g^2*(f + g*x)) - (a*(c^2*f^2 + g^2)^(3/ 2)*ArcTanh[(g - c^2*f*x)/(Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2])])/g^4 + ( b*(c^2*f^2 + g^2)^(3/2)*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqr t[c^2*f^2 + g^2])])/g^4 - (b*(c^2*f^2 + g^2)^(3/2)*ArcSinh[c*x]*Log[1 + (E ^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/g^4 + (b*(c^2*f^2 + g^2)^(3 /2)*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/g^4 - ( b*(c^2*f^2 + g^2)^(3/2)*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^ 2 + g^2]))])/g^4))/Sqrt[1 + c^2*x^2]
3.1.42.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d _) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[Sqrt[d + e*x^2]*( a + b*ArcSinh[c*x])^n, (f + g*x)^m*(d + e*x^2)^(p - 1/2), x], x] /; FreeQ[{ a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IntegerQ[m] && IGtQ[p + 1/2, 0 ] && GtQ[d, 0] && IGtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d _) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[Simp[(d + e*x^2)^p/(1 + c^2*x^2) ^p] Int[(f + g*x)^m*(1 + c^2*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] /; Fre eQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e, c^2*d] && IntegerQ[m] && IntegerQ [p - 1/2] && !GtQ[d, 0]
Time = 0.86 (sec) , antiderivative size = 1557, normalized size of antiderivative = 1.58
method | result | size |
default | \(\text {Expression too large to display}\) | \(1557\) |
parts | \(\text {Expression too large to display}\) | \(1557\) |
a/g*(1/3*((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(3/2)-c ^2*d*f/g*(1/4*(2*c^2*d*(x+f/g)-2*c^2*d*f/g)/c^2/d*((x+f/g)^2*c^2*d-2*c^2*d *f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2)+1/8*(4*c^2*d^2*(c^2*f^2+g^2)/g^2-4 *c^4*d^2*f^2/g^2)/c^2/d*ln((-c^2*d*f/g+c^2*d*(x+f/g))/(c^2*d)^(1/2)+((x+f/ g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2))/(c^2*d)^(1/2))+ d*(c^2*f^2+g^2)/g^2*(((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/ g^2)^(1/2)-c^2*d*f/g*ln((-c^2*d*f/g+c^2*d*(x+f/g))/(c^2*d)^(1/2)+((x+f/g)^ 2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2))/(c^2*d)^(1/2)-d*(c ^2*f^2+g^2)/g^2/(d*(c^2*f^2+g^2)/g^2)^(1/2)*ln((2*d*(c^2*f^2+g^2)/g^2-2*c^ 2*d*f/g*(x+f/g)+2*(d*(c^2*f^2+g^2)/g^2)^(1/2)*((x+f/g)^2*c^2*d-2*c^2*d*f/g *(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2))/(x+f/g))))+b*(c^2*f^2+g^2)^(3/2)*d*(d *(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/g^4*dilog((-(c*x+(c^2*x^2+1)^(1/2))* g-c*f+(c^2*f^2+g^2)^(1/2))/(-c*f+(c^2*f^2+g^2)^(1/2)))-b*(c^2*f^2+g^2)^(3/ 2)*d*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/g^4*dilog(((c*x+(c^2*x^2+1)^( 1/2))*g+c*f+(c^2*f^2+g^2)^(1/2))/(c*f+(c^2*f^2+g^2)^(1/2)))-1/2*b*(d*(c^2* x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*f^3*arcsinh(c*x)^2*c^3*d/g^4-3/4*b*(d*(c^2 *x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*f*arcsinh(c*x)^2*c*d/g^2+1/3*b*(d*(c^2*x^ 2+1))^(1/2)*d/(c^2*x^2+1)/g*arcsinh(c*x)*x^4*c^4-1/9*b*(d*(c^2*x^2+1))^(1/ 2)*d/(c^2*x^2+1)^(1/2)/g*x^3*c^3+5/3*b*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1) /g*arcsinh(c*x)*x^2*c^2-4/3*b*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)^(1/2)...
\[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{g x + f} \,d x } \]
integral((a*c^2*d*x^2 + a*d + (b*c^2*d*x^2 + b*d)*arcsinh(c*x))*sqrt(c^2*d *x^2 + d)/(g*x + f), x)
\[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{f + g x}\, dx \]
Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\text {Exception raised: RuntimeError} \]
Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{3/2}}{f+g\,x} \,d x \]