Integrand size = 30, antiderivative size = 412 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^7} \, dx=-\frac {12 \sqrt {a} b d \sqrt {a+b x^4}}{5 x \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {2 \left (5 a f-9 b d x^2\right ) \sqrt {a+b x^4}}{15 x^3}-\frac {b \left (2 c-3 e x^2\right ) \sqrt {a+b x^4}}{4 x^2}-\frac {\left (2 c+3 e x^2\right ) \left (a+b x^4\right )^{3/2}}{12 x^6}-\frac {\left (3 d-5 f x^2\right ) \left (a+b x^4\right )^{3/2}}{15 x^5}+\frac {1}{2} b^{3/2} c \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {3}{4} \sqrt {a} b e \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )-\frac {12 \sqrt [4]{a} b^{5/4} d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+b x^4}}+\frac {2 \sqrt [4]{a} b^{3/4} \left (9 \sqrt {b} d+5 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{15 \sqrt {a+b x^4}} \] Output:
-12/5*a^(1/2)*b*d*(b*x^4+a)^(1/2)/x/(a^(1/2)+b^(1/2)*x^2)-2/15*(-9*b*d*x^2 +5*a*f)*(b*x^4+a)^(1/2)/x^3-1/4*b*(-3*e*x^2+2*c)*(b*x^4+a)^(1/2)/x^2-1/12* (3*e*x^2+2*c)*(b*x^4+a)^(3/2)/x^6-1/15*(-5*f*x^2+3*d)*(b*x^4+a)^(3/2)/x^5+ 1/2*b^(3/2)*c*arctanh(b^(1/2)*x^2/(b*x^4+a)^(1/2))-3/4*a^(1/2)*b*e*arctanh ((b*x^4+a)^(1/2)/a^(1/2))-12/5*a^(1/4)*b^(5/4)*d*(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)*d+5*a ^(1/2)*f)*(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))/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.16 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.40 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^7} \, dx=\frac {\sqrt {a+b x^4} \left (-5 a^3 c \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},-\frac {b x^4}{a}\right )-6 a^3 d x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {5}{4},-\frac {1}{4},-\frac {b x^4}{a}\right )-10 a^3 f x^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{4},\frac {1}{4},-\frac {b x^4}{a}\right )+3 b e x^6 \left (a+b x^4\right )^2 \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (2,\frac {5}{2},\frac {7}{2},1+\frac {b x^4}{a}\right )\right )}{30 a^2 x^6 \sqrt {1+\frac {b x^4}{a}}} \] Input:
Integrate[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^7,x]
Output:
(Sqrt[a + b*x^4]*(-5*a^3*c*Hypergeometric2F1[-3/2, -3/2, -1/2, -((b*x^4)/a )] - 6*a^3*d*x*Hypergeometric2F1[-3/2, -5/4, -1/4, -((b*x^4)/a)] - 10*a^3* f*x^3*Hypergeometric2F1[-3/2, -3/4, 1/4, -((b*x^4)/a)] + 3*b*e*x^6*(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^6*Sqrt[1 + (b*x^4)/a])
Time = 1.12 (sec) , antiderivative size = 389, normalized size of antiderivative = 0.94, 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.
| \(\displaystyle \int \frac {\left (a+b x^4\right )^{3/2} \left (c+d x+e x^2+f x^3\right )}{x^7} \, dx\) |
| \(\Big \downarrow \) 2364 |
| \(\displaystyle -6 b \int -\frac {\left (20 f x^3+15 e x^2+12 d x+10 c\right ) \sqrt {b x^4+a}}{60 x^3}dx-\frac {1}{60} \left (a+b x^4\right )^{3/2} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} b \int \frac {\left (20 f x^3+15 e x^2+12 d x+10 c\right ) \sqrt {b x^4+a}}{x^3}dx-\frac {1}{60} \left (a+b x^4\right )^{3/2} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}\right )\) |
| \(\Big \downarrow \) 2372 |
| \(\displaystyle \frac {1}{10} b \int \left (\frac {\sqrt {b x^4+a} \left (15 e x^2+10 c\right )}{x^3}+\frac {\left (20 f x^2+12 d\right ) \sqrt {b x^4+a}}{x^2}\right )dx-\frac {1}{60} \left (a+b x^4\right )^{3/2} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{10} b \left (\frac {4 \sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (5 \sqrt {a} f+9 \sqrt {b} d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{b} \sqrt {a+b x^4}}-\frac {24 \sqrt [4]{a} \sqrt [4]{b} d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt {a+b x^4}}+5 \sqrt {b} c \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {15}{2} \sqrt {a} e \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )-\frac {5 \sqrt {a+b x^4} \left (2 c-3 e x^2\right )}{2 x^2}-\frac {4 \sqrt {a+b x^4} \left (9 d-5 f x^2\right )}{3 x}+\frac {24 \sqrt {b} d x \sqrt {a+b x^4}}{\sqrt {a}+\sqrt {b} x^2}\right )-\frac {1}{60} \left (a+b x^4\right )^{3/2} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}\right )\) |
Input:
Int[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^7,x]
Output:
-1/60*(((10*c)/x^6 + (12*d)/x^5 + (15*e)/x^4 + (20*f)/x^3)*(a + b*x^4)^(3/ 2)) + (b*((24*Sqrt[b]*d*x*Sqrt[a + b*x^4])/(Sqrt[a] + Sqrt[b]*x^2) - (5*(2 *c - 3*e*x^2)*Sqrt[a + b*x^4])/(2*x^2) - (4*(9*d - 5*f*x^2)*Sqrt[a + b*x^4 ])/(3*x) + 5*Sqrt[b]*c*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]] - (15*Sqrt[a ]*e*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/2 - (24*a^(1/4)*b^(1/4)*d*(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]*d + 5*Sqrt[a]*f)*(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 = 343, normalized size of antiderivative = 0.83
| method | result | size |
| elliptic | \(-\frac {a c \sqrt {b \,x^{4}+a}}{6 x^{6}}-\frac {a d \sqrt {b \,x^{4}+a}}{5 x^{5}}-\frac {a e \sqrt {b \,x^{4}+a}}{4 x^{4}}-\frac {a f \sqrt {b \,x^{4}+a}}{3 x^{3}}-\frac {2 c b \sqrt {b \,x^{4}+a}}{3 x^{2}}-\frac {7 b d \sqrt {b \,x^{4}+a}}{5 x}+\frac {f b x \sqrt {b \,x^{4}+a}}{3}+\frac {b e \sqrt {b \,x^{4}+a}}{2}+\frac {4 b f a \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {b^{\frac {3}{2}} c \ln \left (2 \sqrt {b}\, x^{2}+2 \sqrt {b \,x^{4}+a}\right )}{2}+\frac {12 i b^{\frac {3}{2}} d \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 \sqrt {a}\, b e \,\operatorname {arctanh}\left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{4}\) | \(343\) |
| default | \(c \left (\frac {b^{\frac {3}{2}} \ln \left (\sqrt {b}\, x^{2}+\sqrt {b \,x^{4}+a}\right )}{2}-\frac {a \sqrt {b \,x^{4}+a}}{6 x^{6}}-\frac {2 b \sqrt {b \,x^{4}+a}}{3 x^{2}}\right )+d \left (-\frac {a \sqrt {b \,x^{4}+a}}{5 x^{5}}-\frac {7 b \sqrt {b \,x^{4}+a}}{5 x}+\frac {12 i b^{\frac {3}{2}} \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+e \left (\frac {b \sqrt {b \,x^{4}+a}}{2}-\frac {a \sqrt {b \,x^{4}+a}}{4 x^{4}}-\frac {3 \sqrt {a}\, b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{4}\right )+f \left (-\frac {a \sqrt {b \,x^{4}+a}}{3 x^{3}}+\frac {b x \sqrt {b \,x^{4}+a}}{3}+\frac {4 a b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(349\) |
| risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (84 x^{5} b d +40 b c \,x^{4}+20 a f \,x^{3}+15 a e \,x^{2}+12 a d x +10 a c \right )}{60 x^{6}}+\frac {4 b f a \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 b \sqrt {a}\, e \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{4}+\frac {12 i b^{\frac {3}{2}} d \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {12 i b^{\frac {3}{2}} d \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {b e \sqrt {b \,x^{4}+a}}{2}+\frac {b^{\frac {3}{2}} c \ln \left (\sqrt {b}\, x^{2}+\sqrt {b \,x^{4}+a}\right )}{2}+\frac {f b x \sqrt {b \,x^{4}+a}}{3}\) | \(364\) |
Input:
int((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^7,x,method=_RETURNVERBOSE)
Output:
-1/6*a*c*(b*x^4+a)^(1/2)/x^6-1/5*a*d*(b*x^4+a)^(1/2)/x^5-1/4*a*e*(b*x^4+a) ^(1/2)/x^4-1/3*a*f*(b*x^4+a)^(1/2)/x^3-2/3*c*b*(b*x^4+a)^(1/2)/x^2-7/5*b*d *(b*x^4+a)^(1/2)/x+1/3*f*b*x*(b*x^4+a)^(1/2)+1/2*b*e*(b*x^4+a)^(1/2)+4/3*b *f*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)+1/2*b^(3/2)*c*ln(2*b^(1/2)*x^2+2*(b*x^4+a)^(1/2))+12/5*I*b^(3/2)*d*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*a^(1/2)*b*e*arctanh(a^( 1/2)/(b*x^4+a)^(1/2))
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^7} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{7}} \,d x } \] Input:
integrate((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^7,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^7, x)
Result contains complex when optimal does not.
Time = 4.90 (sec) , antiderivative size = 406, normalized size of antiderivative = 0.99 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^7} \, dx=\frac {a^{\frac {3}{2}} d \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} + \frac {a^{\frac {3}{2}} f \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} - \frac {\sqrt {a} b c}{2 x^{2} \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} b d \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} - \frac {3 \sqrt {a} b e \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{4} + \frac {\sqrt {a} b f x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} - \frac {a \sqrt {b} c \sqrt {\frac {a}{b x^{4}} + 1}}{6 x^{4}} - \frac {a \sqrt {b} e \sqrt {\frac {a}{b x^{4}} + 1}}{4 x^{2}} + \frac {a \sqrt {b} e}{2 x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {3}{2}} c \sqrt {\frac {a}{b x^{4}} + 1}}{6} + \frac {b^{\frac {3}{2}} c \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{2} + \frac {b^{\frac {3}{2}} e x^{2}}{2 \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{2} c x^{2}}{2 \sqrt {a} \sqrt {1 + \frac {b x^{4}}{a}}} \] Input:
integrate((f*x**3+e*x**2+d*x+c)*(b*x**4+a)**(3/2)/x**7,x)
Output:
a**(3/2)*d*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)*f*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*c/(2*x**2*sq rt(1 + b*x**4/a)) + sqrt(a)*b*d*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*e*asinh(sqrt(a)/(sq rt(b)*x**2))/4 + sqrt(a)*b*f*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), b*x** 4*exp_polar(I*pi)/a)/(4*gamma(5/4)) - a*sqrt(b)*c*sqrt(a/(b*x**4) + 1)/(6* x**4) - a*sqrt(b)*e*sqrt(a/(b*x**4) + 1)/(4*x**2) + a*sqrt(b)*e/(2*x**2*sq rt(a/(b*x**4) + 1)) - b**(3/2)*c*sqrt(a/(b*x**4) + 1)/6 + b**(3/2)*c*asinh (sqrt(b)*x**2/sqrt(a))/2 + b**(3/2)*e*x**2/(2*sqrt(a/(b*x**4) + 1)) - b**2 *c*x**2/(2*sqrt(a)*sqrt(1 + b*x**4/a))
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^7} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{7}} \,d x } \] Input:
integrate((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^7,x, algorithm="maxima")
Output:
-1/12*(3*b^(3/2)*log(-(sqrt(b) - sqrt(b*x^4 + a)/x^2)/(sqrt(b) + sqrt(b*x^ 4 + a)/x^2)) + 6*sqrt(b*x^4 + a)*b/x^2 + 2*(b*x^4 + a)^(3/2)/x^6)*c + inte grate((b*f*x^6 + b*e*x^5 + b*d*x^4 + a*f*x^2 + a*e*x + a*d)*sqrt(b*x^4 + a )/x^6, x)
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^7} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{7}} \,d x } \] Input:
integrate((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^7,x, algorithm="giac")
Output:
integrate((b*x^4 + a)^(3/2)*(f*x^3 + e*x^2 + d*x + c)/x^7, x)
Timed out. \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^7} \, dx=\int \frac {{\left (b\,x^4+a\right )}^{3/2}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x^7} \,d x \] Input:
int(((a + b*x^4)^(3/2)*(c + d*x + e*x^2 + f*x^3))/x^7,x)
Output:
int(((a + b*x^4)^(3/2)*(c + d*x + e*x^2 + f*x^3))/x^7, x)
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^7} \, dx=\frac {-4 \sqrt {b \,x^{4}+a}\, a c -24 \sqrt {b \,x^{4}+a}\, a d x -6 \sqrt {b \,x^{4}+a}\, a e \,x^{2}-40 \sqrt {b \,x^{4}+a}\, a f \,x^{3}-16 \sqrt {b \,x^{4}+a}\, b c \,x^{4}+24 \sqrt {b \,x^{4}+a}\, b d \,x^{5}+12 \sqrt {b \,x^{4}+a}\, b e \,x^{6}+8 \sqrt {b \,x^{4}+a}\, b f \,x^{7}+9 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}-\sqrt {a}\right ) b e \,x^{6}-9 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}+\sqrt {a}\right ) b e \,x^{6}-6 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}-\sqrt {b}\, x^{2}\right ) b c \,x^{6}+6 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}+\sqrt {b}\, x^{2}\right ) b c \,x^{6}-96 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{10}+a \,x^{6}}d x \right ) a^{2} d \,x^{6}-96 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{8}+a \,x^{4}}d x \right ) a^{2} f \,x^{6}}{24 x^{6}} \] Input:
int((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^7,x)
Output:
( - 4*sqrt(a + b*x**4)*a*c - 24*sqrt(a + b*x**4)*a*d*x - 6*sqrt(a + b*x**4 )*a*e*x**2 - 40*sqrt(a + b*x**4)*a*f*x**3 - 16*sqrt(a + b*x**4)*b*c*x**4 + 24*sqrt(a + b*x**4)*b*d*x**5 + 12*sqrt(a + b*x**4)*b*e*x**6 + 8*sqrt(a + b*x**4)*b*f*x**7 + 9*sqrt(a)*log(sqrt(a + b*x**4) - sqrt(a))*b*e*x**6 - 9* sqrt(a)*log(sqrt(a + b*x**4) + sqrt(a))*b*e*x**6 - 6*sqrt(b)*log(sqrt(a + b*x**4) - sqrt(b)*x**2)*b*c*x**6 + 6*sqrt(b)*log(sqrt(a + b*x**4) + sqrt(b )*x**2)*b*c*x**6 - 96*int(sqrt(a + b*x**4)/(a*x**6 + b*x**10),x)*a**2*d*x* *6 - 96*int(sqrt(a + b*x**4)/(a*x**4 + b*x**8),x)*a**2*f*x**6)/(24*x**6)