\(\int \frac {(\pi +c^2 \pi x^2)^{3/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx\) [73]

Optimal result

Integrand size = 26, antiderivative size = 115 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=-\frac {b c \pi ^{3/2}}{6 x^2}-\frac {c^2 \pi \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x}-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {c^3 \pi ^{3/2} (a+b \text {arcsinh}(c x))^2}{2 b}+\frac {4}{3} b c^3 \pi ^{3/2} \log (x) \] Output:

-1/6*b*c*Pi^(3/2)/x^2-c^2*Pi*(Pi*c^2*x^2+Pi)^(1/2)*(a+b*arcsinh(c*x))/x-1/ 
3*(Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(c*x))/x^3+1/2*c^3*Pi^(3/2)*(a+b*arcsi 
nh(c*x))^2/b+4/3*b*c^3*Pi^(3/2)*ln(x)
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.09 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\frac {\pi ^{3/2} \left (-b c x-2 a \sqrt {1+c^2 x^2}-8 a c^2 x^2 \sqrt {1+c^2 x^2}+\left (6 a c^3 x^3-2 b \sqrt {1+c^2 x^2} \left (1+4 c^2 x^2\right )\right ) \text {arcsinh}(c x)+3 b c^3 x^3 \text {arcsinh}(c x)^2+8 b c^3 x^3 \log (c x)\right )}{6 x^3} \] Input:

Integrate[((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^4,x]
 

Output:

(Pi^(3/2)*(-(b*c*x) - 2*a*Sqrt[1 + c^2*x^2] - 8*a*c^2*x^2*Sqrt[1 + c^2*x^2 
] + (6*a*c^3*x^3 - 2*b*Sqrt[1 + c^2*x^2]*(1 + 4*c^2*x^2))*ArcSinh[c*x] + 3 
*b*c^3*x^3*ArcSinh[c*x]^2 + 8*b*c^3*x^3*Log[c*x]))/(6*x^3)
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {6222, 244, 2009, 6220, 14, 6198}

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.

\(\displaystyle \int \frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx\)

\(\Big \downarrow \) 6222

\(\displaystyle \pi c^2 \int \frac {\sqrt {c^2 \pi x^2+\pi } (a+b \text {arcsinh}(c x))}{x^2}dx+\frac {1}{3} \pi ^{3/2} b c \int \frac {c^2 x^2+1}{x^3}dx-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 244

\(\displaystyle \pi c^2 \int \frac {\sqrt {c^2 \pi x^2+\pi } (a+b \text {arcsinh}(c x))}{x^2}dx+\frac {1}{3} \pi ^{3/2} b c \int \left (\frac {c^2}{x}+\frac {1}{x^3}\right )dx-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \pi c^2 \int \frac {\sqrt {c^2 \pi x^2+\pi } (a+b \text {arcsinh}(c x))}{x^2}dx-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {1}{3} \pi ^{3/2} b c \left (c^2 \log (x)-\frac {1}{2 x^2}\right )\)

\(\Big \downarrow \) 6220

\(\displaystyle \pi c^2 \left (\sqrt {\pi } c^2 \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx+\sqrt {\pi } b c \int \frac {1}{x}dx-\frac {\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{x}\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {1}{3} \pi ^{3/2} b c \left (c^2 \log (x)-\frac {1}{2 x^2}\right )\)

\(\Big \downarrow \) 14

\(\displaystyle \pi c^2 \left (\sqrt {\pi } c^2 \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx-\frac {\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{x}+\sqrt {\pi } b c \log (x)\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {1}{3} \pi ^{3/2} b c \left (c^2 \log (x)-\frac {1}{2 x^2}\right )\)

\(\Big \downarrow \) 6198

\(\displaystyle \pi c^2 \left (-\frac {\sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{x}+\frac {\sqrt {\pi } c (a+b \text {arcsinh}(c x))^2}{2 b}+\sqrt {\pi } b c \log (x)\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 x^3}+\frac {1}{3} \pi ^{3/2} b c \left (c^2 \log (x)-\frac {1}{2 x^2}\right )\)

Input:

Int[((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^4,x]
 

Output:

-1/3*((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^3 + (b*c*Pi^(3/2)*(- 
1/2*1/x^2 + c^2*Log[x]))/3 + c^2*Pi*(-((Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSi 
nh[c*x]))/x) + (c*Sqrt[Pi]*(a + b*ArcSinh[c*x])^2)/(2*b) + b*c*Sqrt[Pi]*Lo 
g[x])
 

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6198

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( 
a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c 
^2*d] && NeQ[n, -1]
 

rule 6220

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + 
 (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc 
Sinh[c*x])^n/(f*(m + 1))), x] + (-Simp[b*c*(n/(f*(m + 1)))*Simp[Sqrt[d + e* 
x^2]/Sqrt[1 + c^2*x^2]]   Int[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x 
], x] - Simp[(c^2/(f^2*(m + 1)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] 
Int[(f*x)^(m + 2)*((a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2]), x], x]) /; Fr 
eeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[m, -1]
 

rule 6222

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*Arc 
Sinh[c*x])^n/(f*(m + 1))), x] + (-Simp[2*e*(p/(f^2*(m + 1)))   Int[(f*x)^(m 
 + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*c*(n/(f*( 
m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int[(f*x)^(m + 1)*(1 + c^2*x 
^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e 
, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(621\) vs. \(2(97)=194\).

Time = 1.00 (sec) , antiderivative size = 622, normalized size of antiderivative = 5.41

method	result	size
default	\(-\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{3 \pi \,x^{3}}-\frac {2 a \,c^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{3 \pi x}+\frac {2 a \,c^{4} x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3}+a \,c^{4} \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }+\frac {a \,c^{4} \pi ^{2} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{\sqrt {\pi \,c^{2}}}+\frac {b \,\pi ^{\frac {3}{2}} c^{3} \operatorname {arcsinh}\left (x c \right )^{2}}{2}-\frac {8 b \,\pi ^{\frac {3}{2}} c^{3} \operatorname {arcsinh}\left (x c \right )}{3}+\frac {32 b \,\pi ^{\frac {3}{2}} x^{4} \operatorname {arcsinh}\left (x c \right ) c^{7}}{24 c^{4} x^{4}+9 c^{2} x^{2}+1}-\frac {32 b \,\pi ^{\frac {3}{2}} x^{3} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) c^{6}}{24 c^{4} x^{4}+9 c^{2} x^{2}+1}+\frac {8 b \,\pi ^{\frac {3}{2}} x^{4} c^{7}}{3 \left (24 c^{4} x^{4}+9 c^{2} x^{2}+1\right )}-\frac {8 b \,\pi ^{\frac {3}{2}} x^{2} \left (c^{2} x^{2}+1\right ) c^{5}}{3 \left (24 c^{4} x^{4}+9 c^{2} x^{2}+1\right )}+\frac {12 b \,\pi ^{\frac {3}{2}} x^{2} \operatorname {arcsinh}\left (x c \right ) c^{5}}{24 c^{4} x^{4}+9 c^{2} x^{2}+1}-\frac {20 b \,\pi ^{\frac {3}{2}} x \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) c^{4}}{24 c^{4} x^{4}+9 c^{2} x^{2}+1}-\frac {4 b \,\pi ^{\frac {3}{2}} \left (c^{2} x^{2}+1\right ) c^{3}}{3 \left (24 c^{4} x^{4}+9 c^{2} x^{2}+1\right )}+\frac {4 b \,\pi ^{\frac {3}{2}} \operatorname {arcsinh}\left (x c \right ) c^{3}}{3 \left (24 c^{4} x^{4}+9 c^{2} x^{2}+1\right )}-\frac {13 b \,\pi ^{\frac {3}{2}} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) c^{2}}{3 \left (24 c^{4} x^{4}+9 c^{2} x^{2}+1\right ) x}-\frac {b \,\pi ^{\frac {3}{2}} \left (c^{2} x^{2}+1\right ) c}{6 \left (24 c^{4} x^{4}+9 c^{2} x^{2}+1\right ) x^{2}}-\frac {b \,\pi ^{\frac {3}{2}} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right )}{3 \left (24 c^{4} x^{4}+9 c^{2} x^{2}+1\right ) x^{3}}+\frac {4 b \,\pi ^{\frac {3}{2}} c^{3} \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right )}{3}\)	\(622\)
parts	\(-\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{3 \pi \,x^{3}}-\frac {2 a \,c^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{3 \pi x}+\frac {2 a \,c^{4} x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3}+a \,c^{4} \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }+\frac {a \,c^{4} \pi ^{2} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{\sqrt {\pi \,c^{2}}}+\frac {b \,\pi ^{\frac {3}{2}} c^{3} \operatorname {arcsinh}\left (x c \right )^{2}}{2}-\frac {8 b \,\pi ^{\frac {3}{2}} c^{3} \operatorname {arcsinh}\left (x c \right )}{3}+\frac {32 b \,\pi ^{\frac {3}{2}} x^{4} \operatorname {arcsinh}\left (x c \right ) c^{7}}{24 c^{4} x^{4}+9 c^{2} x^{2}+1}-\frac {32 b \,\pi ^{\frac {3}{2}} x^{3} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) c^{6}}{24 c^{4} x^{4}+9 c^{2} x^{2}+1}+\frac {8 b \,\pi ^{\frac {3}{2}} x^{4} c^{7}}{3 \left (24 c^{4} x^{4}+9 c^{2} x^{2}+1\right )}-\frac {8 b \,\pi ^{\frac {3}{2}} x^{2} \left (c^{2} x^{2}+1\right ) c^{5}}{3 \left (24 c^{4} x^{4}+9 c^{2} x^{2}+1\right )}+\frac {12 b \,\pi ^{\frac {3}{2}} x^{2} \operatorname {arcsinh}\left (x c \right ) c^{5}}{24 c^{4} x^{4}+9 c^{2} x^{2}+1}-\frac {20 b \,\pi ^{\frac {3}{2}} x \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) c^{4}}{24 c^{4} x^{4}+9 c^{2} x^{2}+1}-\frac {4 b \,\pi ^{\frac {3}{2}} \left (c^{2} x^{2}+1\right ) c^{3}}{3 \left (24 c^{4} x^{4}+9 c^{2} x^{2}+1\right )}+\frac {4 b \,\pi ^{\frac {3}{2}} \operatorname {arcsinh}\left (x c \right ) c^{3}}{3 \left (24 c^{4} x^{4}+9 c^{2} x^{2}+1\right )}-\frac {13 b \,\pi ^{\frac {3}{2}} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) c^{2}}{3 \left (24 c^{4} x^{4}+9 c^{2} x^{2}+1\right ) x}-\frac {b \,\pi ^{\frac {3}{2}} \left (c^{2} x^{2}+1\right ) c}{6 \left (24 c^{4} x^{4}+9 c^{2} x^{2}+1\right ) x^{2}}-\frac {b \,\pi ^{\frac {3}{2}} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right )}{3 \left (24 c^{4} x^{4}+9 c^{2} x^{2}+1\right ) x^{3}}+\frac {4 b \,\pi ^{\frac {3}{2}} c^{3} \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right )}{3}\)	\(622\)

Input:

int((Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(x*c))/x^4,x,method=_RETURNVERBOSE)
 

Output:

-1/3*a/Pi/x^3*(Pi*c^2*x^2+Pi)^(5/2)-2/3*a*c^2/Pi/x*(Pi*c^2*x^2+Pi)^(5/2)+2 
/3*a*c^4*x*(Pi*c^2*x^2+Pi)^(3/2)+a*c^4*Pi*x*(Pi*c^2*x^2+Pi)^(1/2)+a*c^4*Pi 
^2*ln(Pi*c^2*x/(Pi*c^2)^(1/2)+(Pi*c^2*x^2+Pi)^(1/2))/(Pi*c^2)^(1/2)+1/2*b* 
Pi^(3/2)*c^3*arcsinh(x*c)^2-8/3*b*Pi^(3/2)*c^3*arcsinh(x*c)+32*b*Pi^(3/2)/ 
(24*c^4*x^4+9*c^2*x^2+1)*x^4*arcsinh(x*c)*c^7-32*b*Pi^(3/2)/(24*c^4*x^4+9* 
c^2*x^2+1)*x^3*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*c^6+8/3*b*Pi^(3/2)/(24*c^4*x 
^4+9*c^2*x^2+1)*x^4*c^7-8/3*b*Pi^(3/2)/(24*c^4*x^4+9*c^2*x^2+1)*x^2*(c^2*x 
^2+1)*c^5+12*b*Pi^(3/2)/(24*c^4*x^4+9*c^2*x^2+1)*x^2*arcsinh(x*c)*c^5-20*b 
*Pi^(3/2)/(24*c^4*x^4+9*c^2*x^2+1)*x*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*c^4-4/ 
3*b*Pi^(3/2)/(24*c^4*x^4+9*c^2*x^2+1)*(c^2*x^2+1)*c^3+4/3*b*Pi^(3/2)/(24*c 
^4*x^4+9*c^2*x^2+1)*arcsinh(x*c)*c^3-13/3*b*Pi^(3/2)/(24*c^4*x^4+9*c^2*x^2 
+1)/x*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*c^2-1/6*b*Pi^(3/2)/(24*c^4*x^4+9*c^2* 
x^2+1)/x^2*(c^2*x^2+1)*c-1/3*b*Pi^(3/2)/(24*c^4*x^4+9*c^2*x^2+1)/x^3*(c^2* 
x^2+1)^(1/2)*arcsinh(x*c)+4/3*b*Pi^(3/2)*c^3*ln((x*c+(c^2*x^2+1)^(1/2))^2- 
1)
 

Fricas [F]

\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\int { \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{4}} \,d x } \] Input:

integrate((pi*c^2*x^2+pi)^(3/2)*(a+b*arcsinh(c*x))/x^4,x, algorithm="frica 
s")
 

Output:

integral(sqrt(pi + pi*c^2*x^2)*(pi*a*c^2*x^2 + pi*a + (pi*b*c^2*x^2 + pi*b 
)*arcsinh(c*x))/x^4, x)
 

Sympy [F]

\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\pi ^{\frac {3}{2}} \left (\int \frac {a \sqrt {c^{2} x^{2} + 1}}{x^{4}}\, dx + \int \frac {a c^{2} \sqrt {c^{2} x^{2} + 1}}{x^{2}}\, dx + \int \frac {b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {b c^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x^{2}}\, dx\right ) \] Input:

integrate((pi*c**2*x**2+pi)**(3/2)*(a+b*asinh(c*x))/x**4,x)
 

Output:

pi**(3/2)*(Integral(a*sqrt(c**2*x**2 + 1)/x**4, x) + Integral(a*c**2*sqrt( 
c**2*x**2 + 1)/x**2, x) + Integral(b*sqrt(c**2*x**2 + 1)*asinh(c*x)/x**4, 
x) + Integral(b*c**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/x**2, x))
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((pi*c^2*x^2+pi)^(3/2)*(a+b*arcsinh(c*x))/x^4,x, algorithm="maxim 
a")
 

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((pi*c^2*x^2+pi)^(3/2)*(a+b*arcsinh(c*x))/x^4,x, algorithm="giac" 
)
 

Output:

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
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}}{x^4} \,d x \] Input:

int(((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(3/2))/x^4,x)
 

Output:

int(((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(3/2))/x^4, x)
 

Reduce [F]

\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\frac {\sqrt {\pi }\, \pi \left (-4 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, a +3 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{4}}d x \right ) b \,x^{3}+3 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{2}}d x \right ) b \,c^{2} x^{3}+3 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a \,c^{3} x^{3}\right )}{3 x^{3}} \] Input:

int((Pi*c^2*x^2+Pi)^(3/2)*(a+b*asinh(c*x))/x^4,x)
 

Output:

(sqrt(pi)*pi*( - 4*sqrt(c**2*x**2 + 1)*a*c**2*x**2 - sqrt(c**2*x**2 + 1)*a 
 + 3*int((sqrt(c**2*x**2 + 1)*asinh(c*x))/x**4,x)*b*x**3 + 3*int((sqrt(c** 
2*x**2 + 1)*asinh(c*x))/x**2,x)*b*c**2*x**3 + 3*log(sqrt(c**2*x**2 + 1) + 
c*x)*a*c**3*x**3))/(3*x**3)