Integrand size = 30, antiderivative size = 430 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^8} \, dx=\frac {4 b c \sqrt {a+b x^4}}{7 x^3}-\frac {12 \sqrt {a} b e \sqrt {a+b x^4}}{5 x \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {6 \left (7 a e+5 b c x^2\right ) \sqrt {a+b x^4}}{35 x^5}-\frac {b \left (2 d-3 f x^2\right ) \sqrt {a+b x^4}}{4 x^2}-\frac {\left (c-7 e x^2\right ) \left (a+b x^4\right )^{3/2}}{7 x^7}-\frac {\left (2 d+3 f x^2\right ) \left (a+b x^4\right )^{3/2}}{12 x^6}+\frac {1}{2} b^{3/2} d \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {3}{4} \sqrt {a} b f \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )-\frac {12 \sqrt [4]{a} b^{5/4} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+b x^4}}+\frac {2 b^{5/4} \left (5 \sqrt {b} c+21 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{35 \sqrt [4]{a} \sqrt {a+b x^4}} \] Output:
4/7*b*c*(b*x^4+a)^(1/2)/x^3-12/5*a^(1/2)*b*e*(b*x^4+a)^(1/2)/x/(a^(1/2)+b^ (1/2)*x^2)-6/35*(5*b*c*x^2+7*a*e)*(b*x^4+a)^(1/2)/x^5-1/4*b*(-3*f*x^2+2*d) *(b*x^4+a)^(1/2)/x^2-1/7*(-7*e*x^2+c)*(b*x^4+a)^(3/2)/x^7-1/12*(3*f*x^2+2* d)*(b*x^4+a)^(3/2)/x^6+1/2*b^(3/2)*d*arctanh(b^(1/2)*x^2/(b*x^4+a)^(1/2))- 3/4*a^(1/2)*b*f*arctanh((b*x^4+a)^(1/2)/a^(1/2))-12/5*a^(1/4)*b^(5/4)*e*(a ^(1/2)+b^(1/2)*x^2)*((b*x^4+a)/(a^(1/2)+b^(1/2)*x^2)^2)^(1/2)*EllipticE(si n(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))/(b*x^4+a)^(1/2)+2/35*b^(5/4)*( 5*b^(1/2)*c+21*a^(1/2)*e)*(a^(1/2)+b^(1/2)*x^2)*((b*x^4+a)/(a^(1/2)+b^(1/2 )*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(b^(1/4)*x/a^(1/4)),1/2*2^(1/2))/a ^(1/4)/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.19 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.38 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^8} \, dx=\frac {\sqrt {a+b x^4} \left (-30 a^3 c \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},-\frac {3}{2},-\frac {3}{4},-\frac {b x^4}{a}\right )+7 x \left (-5 a^3 d \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},-\frac {b x^4}{a}\right )-6 a^3 e x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {5}{4},-\frac {1}{4},-\frac {b x^4}{a}\right )+3 b f x^6 \left (a+b x^4\right )^2 \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (2,\frac {5}{2},\frac {7}{2},1+\frac {b x^4}{a}\right )\right )\right )}{210 a^2 x^7 \sqrt {1+\frac {b x^4}{a}}} \] Input:
Integrate[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^8,x]
Output:
(Sqrt[a + b*x^4]*(-30*a^3*c*Hypergeometric2F1[-7/4, -3/2, -3/4, -((b*x^4)/ a)] + 7*x*(-5*a^3*d*Hypergeometric2F1[-3/2, -3/2, -1/2, -((b*x^4)/a)] - 6* a^3*e*x*Hypergeometric2F1[-3/2, -5/4, -1/4, -((b*x^4)/a)] + 3*b*f*x^6*(a + b*x^4)^2*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[2, 5/2, 7/2, 1 + (b*x^4)/a ])))/(210*a^2*x^7*Sqrt[1 + (b*x^4)/a])
Time = 1.22 (sec) , antiderivative size = 402, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2364, 27, 2372, 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.
| \(\displaystyle \int \frac {\left (a+b x^4\right )^{3/2} \left (c+d x+e x^2+f x^3\right )}{x^8} \, dx\) |
| \(\Big \downarrow \) 2364 |
| \(\displaystyle -6 b \int -\frac {\left (105 f x^3+84 e x^2+70 d x+60 c\right ) \sqrt {b x^4+a}}{420 x^4}dx-\frac {1}{420} \left (a+b x^4\right )^{3/2} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{70} b \int \frac {\left (105 f x^3+84 e x^2+70 d x+60 c\right ) \sqrt {b x^4+a}}{x^4}dx-\frac {1}{420} \left (a+b x^4\right )^{3/2} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right )\) |
| \(\Big \downarrow \) 2372 |
| \(\displaystyle \frac {1}{70} b \int \left (\frac {\sqrt {b x^4+a} \left (84 e x^2+60 c\right )}{x^4}+\frac {\left (105 f x^2+70 d\right ) \sqrt {b x^4+a}}{x^3}\right )dx-\frac {1}{420} \left (a+b x^4\right )^{3/2} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{70} b \left (\frac {4 \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (21 \sqrt {a} e+5 \sqrt {b} c\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a+b x^4}}-\frac {168 \sqrt [4]{a} \sqrt [4]{b} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt {a+b x^4}}+35 \sqrt {b} d \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {105}{2} \sqrt {a} f \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )-\frac {4 \sqrt {a+b x^4} \left (5 c-21 e x^2\right )}{x^3}-\frac {35 \sqrt {a+b x^4} \left (2 d-3 f x^2\right )}{2 x^2}-\frac {168 e \sqrt {a+b x^4}}{x}+\frac {168 \sqrt {b} e x \sqrt {a+b x^4}}{\sqrt {a}+\sqrt {b} x^2}\right )-\frac {1}{420} \left (a+b x^4\right )^{3/2} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right )\) |
Input:
Int[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^8,x]
Output:
-1/420*(((60*c)/x^7 + (70*d)/x^6 + (84*e)/x^5 + (105*f)/x^4)*(a + b*x^4)^( 3/2)) + (b*((-168*e*Sqrt[a + b*x^4])/x + (168*Sqrt[b]*e*x*Sqrt[a + b*x^4]) /(Sqrt[a] + Sqrt[b]*x^2) - (4*(5*c - 21*e*x^2)*Sqrt[a + b*x^4])/x^3 - (35* (2*d - 3*f*x^2)*Sqrt[a + b*x^4])/(2*x^2) + 35*Sqrt[b]*d*ArcTanh[(Sqrt[b]*x ^2)/Sqrt[a + b*x^4]] - (105*Sqrt[a]*f*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/2 - (168*a^(1/4)*b^(1/4)*e*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/Sqrt[a + b*x^4] + (4*b^(1/4)*(5*Sqrt[b]*c + 21*Sqrt[a]*e)*(Sqrt[a] + Sqrt[b]*x^2)* Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x) /a^(1/4)], 1/2])/(a^(1/4)*Sqrt[a + b*x^4])))/70
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Module[{u = IntHide[x^m*Pq, x]}, Simp[u*(a + b*x^n)^p, x] - Simp[b*n*p Int[x^(m + n)*(a + b*x^n)^(p - 1)*ExpandToSum[u/x^(m + 1), x], x], x]] /; FreeQ[{a, b} , x] && PolyQ[Pq, x] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m + Expon[Pq, x] + 1 , 0]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Mo dule[{q = Expon[Pq, x], j, k}, Int[Sum[((c*x)^(m + j)/c^j)*Sum[Coeff[Pq, x, j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2*((q - j)/n) + 1}]*(a + b*x^n)^p, {j, 0, n/2 - 1}], x]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0 ] && !PolyQ[Pq, x^(n/2)]
Result contains complex when optimal does not.
Time = 3.57 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.70
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (588 b e \,x^{6}+280 x^{5} b d +180 b c \,x^{4}+105 a f \,x^{3}+84 a e \,x^{2}+70 a d x +60 a c \right )}{420 x^{7}}+\frac {b \left (\frac {40 c b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {105 \sqrt {a}\, f \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{2}+35 \sqrt {b}\, d \ln \left (\sqrt {b}\, x^{2}+\sqrt {b \,x^{4}+a}\right )+\frac {168 i \sqrt {b}\, e \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+35 f \sqrt {b \,x^{4}+a}\right )}{70}\) | \(299\) |
| elliptic | \(-\frac {a c \sqrt {b \,x^{4}+a}}{7 x^{7}}-\frac {a d \sqrt {b \,x^{4}+a}}{6 x^{6}}-\frac {a e \sqrt {b \,x^{4}+a}}{5 x^{5}}-\frac {a f \sqrt {b \,x^{4}+a}}{4 x^{4}}-\frac {3 b c \sqrt {b \,x^{4}+a}}{7 x^{3}}-\frac {2 b d \sqrt {b \,x^{4}+a}}{3 x^{2}}-\frac {7 b e \sqrt {b \,x^{4}+a}}{5 x}+\frac {f b \sqrt {b \,x^{4}+a}}{2}+\frac {4 b^{2} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {b^{\frac {3}{2}} d \ln \left (2 \sqrt {b}\, x^{2}+2 \sqrt {b \,x^{4}+a}\right )}{2}+\frac {12 i e \,b^{\frac {3}{2}} \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 \sqrt {a}\, b f \,\operatorname {arctanh}\left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{4}\) | \(346\) |
| default | \(c \left (-\frac {a \sqrt {b \,x^{4}+a}}{7 x^{7}}-\frac {3 b \sqrt {b \,x^{4}+a}}{7 x^{3}}+\frac {4 b^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (\frac {b^{\frac {3}{2}} \ln \left (\sqrt {b}\, x^{2}+\sqrt {b \,x^{4}+a}\right )}{2}-\frac {a \sqrt {b \,x^{4}+a}}{6 x^{6}}-\frac {2 b \sqrt {b \,x^{4}+a}}{3 x^{2}}\right )+e \left (-\frac {a \sqrt {b \,x^{4}+a}}{5 x^{5}}-\frac {7 b \sqrt {b \,x^{4}+a}}{5 x}+\frac {12 i b^{\frac {3}{2}} \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+f \left (\frac {b \sqrt {b \,x^{4}+a}}{2}-\frac {a \sqrt {b \,x^{4}+a}}{4 x^{4}}-\frac {3 \sqrt {a}\, b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{4}\right )\) | \(352\) |
Input:
int((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^8,x,method=_RETURNVERBOSE)
Output:
-1/420*(b*x^4+a)^(1/2)*(588*b*e*x^6+280*b*d*x^5+180*b*c*x^4+105*a*f*x^3+84 *a*e*x^2+70*a*d*x+60*a*c)/x^7+1/70*b*(40*c*b/(I/a^(1/2)*b^(1/2))^(1/2)*(1- I/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/ 2)*EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)-105/2*a^(1/2)*f*ln((2*a+2*a^(1 /2)*(b*x^4+a)^(1/2))/x^2)+35*b^(1/2)*d*ln(b^(1/2)*x^2+(b*x^4+a)^(1/2))+168 *I*b^(1/2)*e*a^(1/2)/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*x^2)^( 1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*(EllipticF(x*(I/a^(1/ 2)*b^(1/2))^(1/2),I)-EllipticE(x*(I/a^(1/2)*b^(1/2))^(1/2),I))+35*f*(b*x^4 +a)^(1/2))
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^8} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{8}} \,d x } \] Input:
integrate((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^8,x, algorithm="fricas")
Output:
integral((b*f*x^7 + b*e*x^6 + b*d*x^5 + b*c*x^4 + a*f*x^3 + a*e*x^2 + a*d* x + a*c)*sqrt(b*x^4 + a)/x^8, x)
Result contains complex when optimal does not.
Time = 5.17 (sec) , antiderivative size = 415, normalized size of antiderivative = 0.97 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^8} \, dx=\frac {a^{\frac {3}{2}} c \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} + \frac {a^{\frac {3}{2}} e \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} + \frac {\sqrt {a} b c \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} - \frac {\sqrt {a} b d}{2 x^{2} \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} b e \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} - \frac {3 \sqrt {a} b f \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{4} - \frac {a \sqrt {b} d \sqrt {\frac {a}{b x^{4}} + 1}}{6 x^{4}} - \frac {a \sqrt {b} f \sqrt {\frac {a}{b x^{4}} + 1}}{4 x^{2}} + \frac {a \sqrt {b} f}{2 x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {3}{2}} d \sqrt {\frac {a}{b x^{4}} + 1}}{6} + \frac {b^{\frac {3}{2}} d \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{2} + \frac {b^{\frac {3}{2}} f x^{2}}{2 \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{2} d x^{2}}{2 \sqrt {a} \sqrt {1 + \frac {b x^{4}}{a}}} \] Input:
integrate((f*x**3+e*x**2+d*x+c)*(b*x**4+a)**(3/2)/x**8,x)
Output:
a**(3/2)*c*gamma(-7/4)*hyper((-7/4, -1/2), (-3/4,), b*x**4*exp_polar(I*pi) /a)/(4*x**7*gamma(-3/4)) + a**(3/2)*e*gamma(-5/4)*hyper((-5/4, -1/2), (-1/ 4,), b*x**4*exp_polar(I*pi)/a)/(4*x**5*gamma(-1/4)) + sqrt(a)*b*c*gamma(-3 /4)*hyper((-3/4, -1/2), (1/4,), b*x**4*exp_polar(I*pi)/a)/(4*x**3*gamma(1/ 4)) - sqrt(a)*b*d/(2*x**2*sqrt(1 + b*x**4/a)) + sqrt(a)*b*e*gamma(-1/4)*hy per((-1/2, -1/4), (3/4,), b*x**4*exp_polar(I*pi)/a)/(4*x*gamma(3/4)) - 3*s qrt(a)*b*f*asinh(sqrt(a)/(sqrt(b)*x**2))/4 - a*sqrt(b)*d*sqrt(a/(b*x**4) + 1)/(6*x**4) - a*sqrt(b)*f*sqrt(a/(b*x**4) + 1)/(4*x**2) + a*sqrt(b)*f/(2* x**2*sqrt(a/(b*x**4) + 1)) - b**(3/2)*d*sqrt(a/(b*x**4) + 1)/6 + b**(3/2)* d*asinh(sqrt(b)*x**2/sqrt(a))/2 + b**(3/2)*f*x**2/(2*sqrt(a/(b*x**4) + 1)) - b**2*d*x**2/(2*sqrt(a)*sqrt(1 + b*x**4/a))
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^8} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{8}} \,d x } \] Input:
integrate((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^8,x, algorithm="maxima")
Output:
integrate((b*x^4 + a)^(3/2)*(f*x^3 + e*x^2 + d*x + c)/x^8, x)
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^8} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{8}} \,d x } \] Input:
integrate((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^8,x, algorithm="giac")
Output:
integrate((b*x^4 + a)^(3/2)*(f*x^3 + e*x^2 + d*x + c)/x^8, x)
Timed out. \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^8} \, dx=\int \frac {{\left (b\,x^4+a\right )}^{3/2}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x^8} \,d x \] Input:
int(((a + b*x^4)^(3/2)*(c + d*x + e*x^2 + f*x^3))/x^8,x)
Output:
int(((a + b*x^4)^(3/2)*(c + d*x + e*x^2 + f*x^3))/x^8, x)
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^8} \, dx=\frac {24 \sqrt {b \,x^{4}+a}\, a c -20 \sqrt {b \,x^{4}+a}\, a d x -120 \sqrt {b \,x^{4}+a}\, a e \,x^{2}-30 \sqrt {b \,x^{4}+a}\, a f \,x^{3}-120 \sqrt {b \,x^{4}+a}\, b c \,x^{4}-80 \sqrt {b \,x^{4}+a}\, b d \,x^{5}+120 \sqrt {b \,x^{4}+a}\, b e \,x^{6}+60 \sqrt {b \,x^{4}+a}\, b f \,x^{7}+45 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}-\sqrt {a}\right ) b f \,x^{7}-45 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}+\sqrt {a}\right ) b f \,x^{7}-30 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}-\sqrt {b}\, x^{2}\right ) b d \,x^{7}+30 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}+\sqrt {b}\, x^{2}\right ) b d \,x^{7}+288 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{12}+a \,x^{8}}d x \right ) a^{2} c \,x^{7}-480 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{10}+a \,x^{6}}d x \right ) a^{2} e \,x^{7}}{120 x^{7}} \] Input:
int((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^8,x)
Output:
(24*sqrt(a + b*x**4)*a*c - 20*sqrt(a + b*x**4)*a*d*x - 120*sqrt(a + b*x**4 )*a*e*x**2 - 30*sqrt(a + b*x**4)*a*f*x**3 - 120*sqrt(a + b*x**4)*b*c*x**4 - 80*sqrt(a + b*x**4)*b*d*x**5 + 120*sqrt(a + b*x**4)*b*e*x**6 + 60*sqrt(a + b*x**4)*b*f*x**7 + 45*sqrt(a)*log(sqrt(a + b*x**4) - sqrt(a))*b*f*x**7 - 45*sqrt(a)*log(sqrt(a + b*x**4) + sqrt(a))*b*f*x**7 - 30*sqrt(b)*log(sqr t(a + b*x**4) - sqrt(b)*x**2)*b*d*x**7 + 30*sqrt(b)*log(sqrt(a + b*x**4) + sqrt(b)*x**2)*b*d*x**7 + 288*int(sqrt(a + b*x**4)/(a*x**8 + b*x**12),x)*a **2*c*x**7 - 480*int(sqrt(a + b*x**4)/(a*x**6 + b*x**10),x)*a**2*e*x**7)/( 120*x**7)