Integrand size = 30, antiderivative size = 405 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^6} \, dx=-\frac {12 \sqrt {a} b c \sqrt {a+b x^4}}{5 x \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {2 \left (5 a e-9 b c x^2\right ) \sqrt {a+b x^4}}{15 x^3}+\frac {3}{4} b \left (d+f x^2\right ) \sqrt {a+b x^4}-\frac {\left (3 c-5 e x^2\right ) \left (a+b x^4\right )^{3/2}}{15 x^5}-\frac {\left (d+2 f x^2\right ) \left (a+b x^4\right )^{3/2}}{4 x^4}+\frac {3}{4} a \sqrt {b} f \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {3}{4} \sqrt {a} b d \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )-\frac {12 \sqrt [4]{a} b^{5/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+b x^4}}+\frac {2 \sqrt [4]{a} b^{3/4} \left (9 \sqrt {b} c+5 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{15 \sqrt {a+b x^4}} \] Output:
-12/5*a^(1/2)*b*c*(b*x^4+a)^(1/2)/x/(a^(1/2)+b^(1/2)*x^2)-2/15*(-9*b*c*x^2 +5*a*e)*(b*x^4+a)^(1/2)/x^3+3/4*b*(f*x^2+d)*(b*x^4+a)^(1/2)-1/15*(-5*e*x^2 +3*c)*(b*x^4+a)^(3/2)/x^5-1/4*(2*f*x^2+d)*(b*x^4+a)^(3/2)/x^4+3/4*a*b^(1/2 )*f*arctanh(b^(1/2)*x^2/(b*x^4+a)^(1/2))-3/4*a^(1/2)*b*d*arctanh((b*x^4+a) ^(1/2)/a^(1/2))-12/5*a^(1/4)*b^(5/4)*c*(a^(1/2)+b^(1/2)*x^2)*((b*x^4+a)/(a ^(1/2)+b^(1/2)*x^2)^2)^(1/2)*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/ 2*2^(1/2))/(b*x^4+a)^(1/2)+2/15*a^(1/4)*b^(3/4)*(9*b^(1/2)*c+5*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)*InverseJac obiAM(2*arctan(b^(1/4)*x/a^(1/4)),1/2*2^(1/2))/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.17 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.41 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^6} \, dx=\frac {\sqrt {a+b x^4} \left (-6 a^3 c \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {5}{4},-\frac {1}{4},-\frac {b x^4}{a}\right )-10 a^3 e x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{4},\frac {1}{4},-\frac {b x^4}{a}\right )-15 a^3 f x^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {1}{2},\frac {1}{2},-\frac {b x^4}{a}\right )+3 b d x^5 \left (a+b x^4\right )^2 \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (2,\frac {5}{2},\frac {7}{2},1+\frac {b x^4}{a}\right )\right )}{30 a^2 x^5 \sqrt {1+\frac {b x^4}{a}}} \] Input:
Integrate[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^6,x]
Output:
(Sqrt[a + b*x^4]*(-6*a^3*c*Hypergeometric2F1[-3/2, -5/4, -1/4, -((b*x^4)/a )] - 10*a^3*e*x^2*Hypergeometric2F1[-3/2, -3/4, 1/4, -((b*x^4)/a)] - 15*a^ 3*f*x^3*Hypergeometric2F1[-3/2, -1/2, 1/2, -((b*x^4)/a)] + 3*b*d*x^5*(a + b*x^4)^2*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[2, 5/2, 7/2, 1 + (b*x^4)/a] ))/(30*a^2*x^5*Sqrt[1 + (b*x^4)/a])
Time = 1.11 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.95, 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.
| \(\displaystyle \int \frac {\left (a+b x^4\right )^{3/2} \left (c+d x+e x^2+f x^3\right )}{x^6} \, dx\) |
| \(\Big \downarrow \) 2364 |
| \(\displaystyle -6 b \int -\frac {\left (30 f x^3+20 e x^2+15 d x+12 c\right ) \sqrt {b x^4+a}}{60 x^2}dx-\frac {1}{60} \left (a+b x^4\right )^{3/2} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} b \int \frac {\left (30 f x^3+20 e x^2+15 d x+12 c\right ) \sqrt {b x^4+a}}{x^2}dx-\frac {1}{60} \left (a+b x^4\right )^{3/2} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}\right )\) |
| \(\Big \downarrow \) 2372 |
| \(\displaystyle \frac {1}{10} b \int \left (\frac {\sqrt {b x^4+a} \left (20 e x^2+12 c\right )}{x^2}+\frac {\left (30 f x^2+15 d\right ) \sqrt {b x^4+a}}{x}\right )dx-\frac {1}{60} \left (a+b x^4\right )^{3/2} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{10} b \left (\frac {4 \sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (5 \sqrt {a} e+9 \sqrt {b} c\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{b} \sqrt {a+b x^4}}-\frac {24 \sqrt [4]{a} \sqrt [4]{b} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt {a+b x^4}}-\frac {15}{2} \sqrt {a} d \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )+\frac {15 a f \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{2 \sqrt {b}}-\frac {4 \sqrt {a+b x^4} \left (9 c-5 e x^2\right )}{3 x}+\frac {24 \sqrt {b} c x \sqrt {a+b x^4}}{\sqrt {a}+\sqrt {b} x^2}+\frac {15}{2} \sqrt {a+b x^4} \left (d+f x^2\right )\right )-\frac {1}{60} \left (a+b x^4\right )^{3/2} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}\right )\) |
Input:
Int[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^6,x]
Output:
-1/60*(((12*c)/x^5 + (15*d)/x^4 + (20*e)/x^3 + (30*f)/x^2)*(a + b*x^4)^(3/ 2)) + (b*((24*Sqrt[b]*c*x*Sqrt[a + b*x^4])/(Sqrt[a] + Sqrt[b]*x^2) - (4*(9 *c - 5*e*x^2)*Sqrt[a + b*x^4])/(3*x) + (15*(d + f*x^2)*Sqrt[a + b*x^4])/2 + (15*a*f*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]])/(2*Sqrt[b]) - (15*Sqrt[a ]*d*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/2 - (24*a^(1/4)*b^(1/4)*c*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticE[2*ArcT an[(b^(1/4)*x)/a^(1/4)], 1/2])/Sqrt[a + b*x^4] + (4*a^(1/4)*(9*Sqrt[b]*c + 5*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])/(3*b^(1/4)*Sqrt[a + b*x^4])))/10
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 = 3.22 (sec) , antiderivative size = 344, normalized size of antiderivative = 0.85
| method | result | size |
| elliptic | \(-\frac {a c \sqrt {b \,x^{4}+a}}{5 x^{5}}-\frac {a d \sqrt {b \,x^{4}+a}}{4 x^{4}}-\frac {a e \sqrt {b \,x^{4}+a}}{3 x^{3}}-\frac {a f \sqrt {b \,x^{4}+a}}{2 x^{2}}-\frac {7 c b \sqrt {b \,x^{4}+a}}{5 x}+\frac {f b \,x^{2} \sqrt {b \,x^{4}+a}}{4}+\frac {b e x \sqrt {b \,x^{4}+a}}{3}+\frac {d \sqrt {b \,x^{4}+a}\, b}{2}+\frac {4 b e a \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {3 a \sqrt {b}\, f \ln \left (2 \sqrt {b}\, x^{2}+2 \sqrt {b \,x^{4}+a}\right )}{4}+\frac {12 i b^{\frac {3}{2}} c \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 d \sqrt {a}\, b \,\operatorname {arctanh}\left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{4}\) | \(344\) |
| default | \(c \left (-\frac {a \sqrt {b \,x^{4}+a}}{5 x^{5}}-\frac {7 b \sqrt {b \,x^{4}+a}}{5 x}+\frac {12 i b^{\frac {3}{2}} \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (\frac {b \sqrt {b \,x^{4}+a}}{2}-\frac {a \sqrt {b \,x^{4}+a}}{4 x^{4}}-\frac {3 \sqrt {a}\, b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{4}\right )+e \left (-\frac {a \sqrt {b \,x^{4}+a}}{3 x^{3}}+\frac {b x \sqrt {b \,x^{4}+a}}{3}+\frac {4 a b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+f \left (-\frac {a \sqrt {b \,x^{4}+a}}{2 x^{2}}+\frac {b \,x^{2} \sqrt {b \,x^{4}+a}}{4}+\frac {3 a \sqrt {b}\, \ln \left (\sqrt {b}\, x^{2}+\sqrt {b \,x^{4}+a}\right )}{4}\right )\) | \(350\) |
| risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (84 b c \,x^{4}+30 a f \,x^{3}+20 a e \,x^{2}+15 a d x +12 a c \right )}{60 x^{5}}+\frac {4 b e a \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 b \sqrt {a}\, d \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{4}+\frac {3 \sqrt {b}\, a f \ln \left (\sqrt {b}\, x^{2}+\sqrt {b \,x^{4}+a}\right )}{4}+\frac {d \sqrt {b \,x^{4}+a}\, b}{2}+\frac {b e x \sqrt {b \,x^{4}+a}}{3}+\frac {12 i b^{\frac {3}{2}} c \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {12 i b^{\frac {3}{2}} c \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {f b \,x^{2} \sqrt {b \,x^{4}+a}}{4}\) | \(374\) |
Input:
int((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^6,x,method=_RETURNVERBOSE)
Output:
-1/5*a*c*(b*x^4+a)^(1/2)/x^5-1/4*a*d*(b*x^4+a)^(1/2)/x^4-1/3*a*e*(b*x^4+a) ^(1/2)/x^3-1/2*a*f*(b*x^4+a)^(1/2)/x^2-7/5*c*b*(b*x^4+a)^(1/2)/x+1/4*f*b*x ^2*(b*x^4+a)^(1/2)+1/3*b*e*x*(b*x^4+a)^(1/2)+1/2*d*(b*x^4+a)^(1/2)*b+4/3*b *e*a/(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)+3/4*a*b^(1/2)*f*ln(2*b^(1/2)*x^2+2*(b*x^4+a)^(1/2))+12/5*I*b^(3/2)*c*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))-3/4*d*a^(1/2)*b*arctanh(a ^(1/2)/(b*x^4+a)^(1/2))
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^6} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{6}} \,d x } \] Input:
integrate((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^6,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^6, x)
Result contains complex when optimal does not.
Time = 5.39 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.95 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^6} \, dx=\frac {a^{\frac {3}{2}} c \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} + \frac {a^{\frac {3}{2}} e \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} - \frac {a^{\frac {3}{2}} f}{2 x^{2} \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} b c \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} - \frac {3 \sqrt {a} b d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{4} + \frac {\sqrt {a} b e x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {\sqrt {a} b f x^{2} \sqrt {1 + \frac {b x^{4}}{a}}}{4} - \frac {\sqrt {a} b f x^{2}}{2 \sqrt {1 + \frac {b x^{4}}{a}}} - \frac {a \sqrt {b} d \sqrt {\frac {a}{b x^{4}} + 1}}{4 x^{2}} + \frac {a \sqrt {b} d}{2 x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} + \frac {3 a \sqrt {b} f \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{4} + \frac {b^{\frac {3}{2}} d x^{2}}{2 \sqrt {\frac {a}{b x^{4}} + 1}} \] Input:
integrate((f*x**3+e*x**2+d*x+c)*(b*x**4+a)**(3/2)/x**6,x)
Output:
a**(3/2)*c*gamma(-5/4)*hyper((-5/4, -1/2), (-1/4,), b*x**4*exp_polar(I*pi) /a)/(4*x**5*gamma(-1/4)) + a**(3/2)*e*gamma(-3/4)*hyper((-3/4, -1/2), (1/4 ,), b*x**4*exp_polar(I*pi)/a)/(4*x**3*gamma(1/4)) - a**(3/2)*f/(2*x**2*sqr t(1 + b*x**4/a)) + sqrt(a)*b*c*gamma(-1/4)*hyper((-1/2, -1/4), (3/4,), b*x **4*exp_polar(I*pi)/a)/(4*x*gamma(3/4)) - 3*sqrt(a)*b*d*asinh(sqrt(a)/(sqr t(b)*x**2))/4 + sqrt(a)*b*e*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), b*x**4 *exp_polar(I*pi)/a)/(4*gamma(5/4)) + sqrt(a)*b*f*x**2*sqrt(1 + b*x**4/a)/4 - sqrt(a)*b*f*x**2/(2*sqrt(1 + b*x**4/a)) - a*sqrt(b)*d*sqrt(a/(b*x**4) + 1)/(4*x**2) + a*sqrt(b)*d/(2*x**2*sqrt(a/(b*x**4) + 1)) + 3*a*sqrt(b)*f*a sinh(sqrt(b)*x**2/sqrt(a))/4 + b**(3/2)*d*x**2/(2*sqrt(a/(b*x**4) + 1))
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^6} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{6}} \,d x } \] Input:
integrate((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^6,x, algorithm="maxima")
Output:
integrate((b*x^4 + a)^(3/2)*(f*x^3 + e*x^2 + d*x + c)/x^6, x)
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^6} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{6}} \,d x } \] Input:
integrate((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^6,x, algorithm="giac")
Output:
integrate((b*x^4 + a)^(3/2)*(f*x^3 + e*x^2 + d*x + c)/x^6, x)
Timed out. \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^6} \, dx=\int \frac {{\left (b\,x^4+a\right )}^{3/2}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x^6} \,d x \] Input:
int(((a + b*x^4)^(3/2)*(c + d*x + e*x^2 + f*x^3))/x^6,x)
Output:
int(((a + b*x^4)^(3/2)*(c + d*x + e*x^2 + f*x^3))/x^6, x)
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^6} \, dx=\frac {-24 \sqrt {b \,x^{4}+a}\, a c -6 \sqrt {b \,x^{4}+a}\, a d x -40 \sqrt {b \,x^{4}+a}\, a e \,x^{2}-12 \sqrt {b \,x^{4}+a}\, a f \,x^{3}+24 \sqrt {b \,x^{4}+a}\, b c \,x^{4}+12 \sqrt {b \,x^{4}+a}\, b d \,x^{5}+8 \sqrt {b \,x^{4}+a}\, b e \,x^{6}+6 \sqrt {b \,x^{4}+a}\, b f \,x^{7}+9 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}-\sqrt {a}\right ) b d \,x^{5}-9 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}+\sqrt {a}\right ) b d \,x^{5}-9 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}-\sqrt {b}\, x^{2}\right ) a f \,x^{5}+9 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}+\sqrt {b}\, x^{2}\right ) a f \,x^{5}-96 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{10}+a \,x^{6}}d x \right ) a^{2} c \,x^{5}-96 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{8}+a \,x^{4}}d x \right ) a^{2} e \,x^{5}}{24 x^{5}} \] Input:
int((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^6,x)
Output:
( - 24*sqrt(a + b*x**4)*a*c - 6*sqrt(a + b*x**4)*a*d*x - 40*sqrt(a + b*x** 4)*a*e*x**2 - 12*sqrt(a + b*x**4)*a*f*x**3 + 24*sqrt(a + b*x**4)*b*c*x**4 + 12*sqrt(a + b*x**4)*b*d*x**5 + 8*sqrt(a + b*x**4)*b*e*x**6 + 6*sqrt(a + b*x**4)*b*f*x**7 + 9*sqrt(a)*log(sqrt(a + b*x**4) - sqrt(a))*b*d*x**5 - 9* sqrt(a)*log(sqrt(a + b*x**4) + sqrt(a))*b*d*x**5 - 9*sqrt(b)*log(sqrt(a + b*x**4) - sqrt(b)*x**2)*a*f*x**5 + 9*sqrt(b)*log(sqrt(a + b*x**4) + sqrt(b )*x**2)*a*f*x**5 - 96*int(sqrt(a + b*x**4)/(a*x**6 + b*x**10),x)*a**2*c*x* *5 - 96*int(sqrt(a + b*x**4)/(a*x**4 + b*x**8),x)*a**2*e*x**5)/(24*x**5)