Integrand size = 22, antiderivative size = 610 \[ \int \frac {A+B x^3}{x^5 \left (a+b x^3\right )^{5/2}} \, dx=-\frac {A}{4 a x^4 \left (a+b x^3\right )^{3/2}}-\frac {17 A b-8 a B}{36 a^2 x \left (a+b x^3\right )^{3/2}}-\frac {11 (17 A b-8 a B)}{108 a^3 x \sqrt {a+b x^3}}+\frac {55 (17 A b-8 a B) \sqrt {a+b x^3}}{216 a^4 x}-\frac {55 \sqrt [3]{b} (17 A b-8 a B) \sqrt {a+b x^3}}{216 a^4 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {55 \sqrt {2-\sqrt {3}} \sqrt [3]{b} (17 A b-8 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{144\ 3^{3/4} a^{11/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {55 \sqrt [3]{b} (17 A b-8 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{108 \sqrt {2} \sqrt [4]{3} a^{11/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
-1/4*A/a/x^4/(b*x^3+a)^(3/2)-1/36*(17*A*b-8*B*a)/a^2/x/(b*x^3+a)^(3/2)-11/ 108*(17*A*b-8*B*a)/a^3/x/(b*x^3+a)^(1/2)+55/216*(17*A*b-8*B*a)*(b*x^3+a)^( 1/2)/a^4/x-55/216*b^(1/3)*(17*A*b-8*B*a)*(b*x^3+a)^(1/2)/a^4/((1+3^(1/2))* a^(1/3)+b^(1/3)*x)+55/432*(1/2*6^(1/2)-1/2*2^(1/2))*b^(1/3)*(17*A*b-8*B*a) *(a^(1/3)+b^(1/3)*x)*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/((1+3^(1/2)) *a^(1/3)+b^(1/3)*x)^2)^(1/2)*EllipticE(((1-3^(1/2))*a^(1/3)+b^(1/3)*x)/((1 +3^(1/2))*a^(1/3)+b^(1/3)*x),I*3^(1/2)+2*I)*3^(1/4)/a^(11/3)/(a^(1/3)*(a^( 1/3)+b^(1/3)*x)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)^2)^(1/2)/(b*x^3+a)^(1/2)-5 5/648*b^(1/3)*(17*A*b-8*B*a)*(a^(1/3)+b^(1/3)*x)*((a^(2/3)-a^(1/3)*b^(1/3) *x+b^(2/3)*x^2)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)^2)^(1/2)*EllipticF(((1-3^( 1/2))*a^(1/3)+b^(1/3)*x)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x),I*3^(1/2)+2*I)*2^ (1/2)*3^(3/4)/a^(11/3)/(a^(1/3)*(a^(1/3)+b^(1/3)*x)/((1+3^(1/2))*a^(1/3)+b ^(1/3)*x)^2)^(1/2)/(b*x^3+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.05 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.14 \[ \int \frac {A+B x^3}{x^5 \left (a+b x^3\right )^{5/2}} \, dx=\frac {-a^2 A+\left (\frac {17 A b}{2}-4 a B\right ) x^3 \left (a+b x^3\right ) \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {5}{2},\frac {2}{3},-\frac {b x^3}{a}\right )}{4 a^3 x^4 \left (a+b x^3\right )^{3/2}} \] Input:
Integrate[(A + B*x^3)/(x^5*(a + b*x^3)^(5/2)),x]
Output:
(-(a^2*A) + ((17*A*b)/2 - 4*a*B)*x^3*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hyper geometric2F1[-1/3, 5/2, 2/3, -((b*x^3)/a)])/(4*a^3*x^4*(a + b*x^3)^(3/2))
Time = 1.05 (sec) , antiderivative size = 609, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {955, 819, 819, 847, 832, 759, 2416}
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.
| \(\displaystyle \int \frac {A+B x^3}{x^5 \left (a+b x^3\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 955 |
| \(\displaystyle -\frac {(17 A b-8 a B) \int \frac {1}{x^2 \left (b x^3+a\right )^{5/2}}dx}{8 a}-\frac {A}{4 a x^4 \left (a+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle -\frac {(17 A b-8 a B) \left (\frac {11 \int \frac {1}{x^2 \left (b x^3+a\right )^{3/2}}dx}{9 a}+\frac {2}{9 a x \left (a+b x^3\right )^{3/2}}\right )}{8 a}-\frac {A}{4 a x^4 \left (a+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle -\frac {(17 A b-8 a B) \left (\frac {11 \left (\frac {5 \int \frac {1}{x^2 \sqrt {b x^3+a}}dx}{3 a}+\frac {2}{3 a x \sqrt {a+b x^3}}\right )}{9 a}+\frac {2}{9 a x \left (a+b x^3\right )^{3/2}}\right )}{8 a}-\frac {A}{4 a x^4 \left (a+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle -\frac {(17 A b-8 a B) \left (\frac {11 \left (\frac {5 \left (\frac {b \int \frac {x}{\sqrt {b x^3+a}}dx}{2 a}-\frac {\sqrt {a+b x^3}}{a x}\right )}{3 a}+\frac {2}{3 a x \sqrt {a+b x^3}}\right )}{9 a}+\frac {2}{9 a x \left (a+b x^3\right )^{3/2}}\right )}{8 a}-\frac {A}{4 a x^4 \left (a+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 832 |
| \(\displaystyle -\frac {(17 A b-8 a B) \left (\frac {11 \left (\frac {5 \left (\frac {b \left (\frac {\int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}\right )}{2 a}-\frac {\sqrt {a+b x^3}}{a x}\right )}{3 a}+\frac {2}{3 a x \sqrt {a+b x^3}}\right )}{9 a}+\frac {2}{9 a x \left (a+b x^3\right )^{3/2}}\right )}{8 a}-\frac {A}{4 a x^4 \left (a+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle -\frac {(17 A b-8 a B) \left (\frac {11 \left (\frac {5 \left (\frac {b \left (\frac {\int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )}{2 a}-\frac {\sqrt {a+b x^3}}{a x}\right )}{3 a}+\frac {2}{3 a x \sqrt {a+b x^3}}\right )}{9 a}+\frac {2}{9 a x \left (a+b x^3\right )^{3/2}}\right )}{8 a}-\frac {A}{4 a x^4 \left (a+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle -\frac {(17 A b-8 a B) \left (\frac {11 \left (\frac {5 \left (\frac {b \left (\frac {\frac {2 \sqrt {a+b x^3}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )}{2 a}-\frac {\sqrt {a+b x^3}}{a x}\right )}{3 a}+\frac {2}{3 a x \sqrt {a+b x^3}}\right )}{9 a}+\frac {2}{9 a x \left (a+b x^3\right )^{3/2}}\right )}{8 a}-\frac {A}{4 a x^4 \left (a+b x^3\right )^{3/2}}\) |
Input:
Int[(A + B*x^3)/(x^5*(a + b*x^3)^(5/2)),x]
Output:
-1/4*A/(a*x^4*(a + b*x^3)^(3/2)) - ((17*A*b - 8*a*B)*(2/(9*a*x*(a + b*x^3) ^(3/2)) + (11*(2/(3*a*x*Sqrt[a + b*x^3]) + (5*(-(Sqrt[a + b*x^3]/(a*x)) + (b*(((2*Sqrt[a + b*x^3])/(b^(1/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) - ( 3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(1/3)*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^ (1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Elli pticE[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(b^(1/3)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*x) )/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b*x^3]))/b^(1/3) - (2*(1 - Sqrt[3])*Sqrt[2 + Sqrt[3]]*a^(1/3)*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]* EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3 ) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*b^(2/3)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b*x^3])))/(2 *a)))/(3*a)))/(9*a)))/(8*a)
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* ((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & & PosQ[a]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 1) + 1)/(a*n*(p + 1)) Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a , b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] ], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 - Sqrt[3]))*(s/r) Int[1/Sqrt[a + b*x ^3], x], x] + Simp[1/r Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x ]] /; FreeQ[{a, b}, x] && PosQ[a]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)) Int[(e *x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b* c - a*d, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) ]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( (1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt [3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) *s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
Time = 3.23 (sec) , antiderivative size = 581, normalized size of antiderivative = 0.95
| method | result | size |
| elliptic | \(\frac {2 x^{2} \left (A b -B a \right ) \sqrt {b \,x^{3}+a}}{9 a^{3} b \left (x^{3}+\frac {a}{b}\right )^{2}}+\frac {2 b \,x^{2} \left (23 A b -14 B a \right )}{27 a^{4} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}-\frac {A \sqrt {b \,x^{3}+a}}{4 a^{3} x^{4}}+\frac {\left (21 A b -8 B a \right ) \sqrt {b \,x^{3}+a}}{8 a^{4} x}-\frac {2 i \left (-\frac {b \left (23 A b -14 B a \right )}{27 a^{4}}-\frac {b \left (21 A b -8 B a \right )}{16 a^{4}}\right ) \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )+\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{b}\right )}{3 b \sqrt {b \,x^{3}+a}}\) | \(581\) |
| default | \(\text {Expression too large to display}\) | \(1034\) |
| risch | \(\text {Expression too large to display}\) | \(1462\) |
Input:
int((B*x^3+A)/x^5/(b*x^3+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/9*x^2/a^3/b*(A*b-B*a)*(b*x^3+a)^(1/2)/(x^3+a/b)^2+2/27*b*x^2/a^4*(23*A*b -14*B*a)/((x^3+a/b)*b)^(1/2)-1/4/a^3*A*(b*x^3+a)^(1/2)/x^4+1/8/a^4*(21*A*b -8*B*a)*(b*x^3+a)^(1/2)/x-2/3*I*(-1/27*b/a^4*(23*A*b-14*B*a)-1/16*b*(21*A* b-8*B*a)/a^4)*3^(1/2)/b*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^ (1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^( 1/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x +1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^( 1/3))^(1/2)/(b*x^3+a)^(1/2)*((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^ 2)^(1/3))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b *(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/ 3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))+1/b*(-a* b^2)^(1/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/ b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1 /3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)))
Time = 0.11 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.30 \[ \int \frac {A+B x^3}{x^5 \left (a+b x^3\right )^{5/2}} \, dx=-\frac {55 \, {\left ({\left (8 \, B a b^{2} - 17 \, A b^{3}\right )} x^{10} + 2 \, {\left (8 \, B a^{2} b - 17 \, A a b^{2}\right )} x^{7} + {\left (8 \, B a^{3} - 17 \, A a^{2} b\right )} x^{4}\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (55 \, {\left (8 \, B a b^{2} - 17 \, A b^{3}\right )} x^{9} + 88 \, {\left (8 \, B a^{2} b - 17 \, A a b^{2}\right )} x^{6} + 54 \, A a^{3} + 27 \, {\left (8 \, B a^{3} - 17 \, A a^{2} b\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{216 \, {\left (a^{4} b^{2} x^{10} + 2 \, a^{5} b x^{7} + a^{6} x^{4}\right )}} \] Input:
integrate((B*x^3+A)/x^5/(b*x^3+a)^(5/2),x, algorithm="fricas")
Output:
-1/216*(55*((8*B*a*b^2 - 17*A*b^3)*x^10 + 2*(8*B*a^2*b - 17*A*a*b^2)*x^7 + (8*B*a^3 - 17*A*a^2*b)*x^4)*sqrt(b)*weierstrassZeta(0, -4*a/b, weierstras sPInverse(0, -4*a/b, x)) + (55*(8*B*a*b^2 - 17*A*b^3)*x^9 + 88*(8*B*a^2*b - 17*A*a*b^2)*x^6 + 54*A*a^3 + 27*(8*B*a^3 - 17*A*a^2*b)*x^3)*sqrt(b*x^3 + a))/(a^4*b^2*x^10 + 2*a^5*b*x^7 + a^6*x^4)
Time = 165.61 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.14 \[ \int \frac {A+B x^3}{x^5 \left (a+b x^3\right )^{5/2}} \, dx=\frac {A \Gamma \left (- \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, \frac {5}{2} \\ - \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {5}{2}} x^{4} \Gamma \left (- \frac {1}{3}\right )} + \frac {B \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {5}{2} \\ \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {5}{2}} x \Gamma \left (\frac {2}{3}\right )} \] Input:
integrate((B*x**3+A)/x**5/(b*x**3+a)**(5/2),x)
Output:
A*gamma(-4/3)*hyper((-4/3, 5/2), (-1/3,), b*x**3*exp_polar(I*pi)/a)/(3*a** (5/2)*x**4*gamma(-1/3)) + B*gamma(-1/3)*hyper((-1/3, 5/2), (2/3,), b*x**3* exp_polar(I*pi)/a)/(3*a**(5/2)*x*gamma(2/3))
\[ \int \frac {A+B x^3}{x^5 \left (a+b x^3\right )^{5/2}} \, dx=\int { \frac {B x^{3} + A}{{\left (b x^{3} + a\right )}^{\frac {5}{2}} x^{5}} \,d x } \] Input:
integrate((B*x^3+A)/x^5/(b*x^3+a)^(5/2),x, algorithm="maxima")
Output:
integrate((B*x^3 + A)/((b*x^3 + a)^(5/2)*x^5), x)
\[ \int \frac {A+B x^3}{x^5 \left (a+b x^3\right )^{5/2}} \, dx=\int { \frac {B x^{3} + A}{{\left (b x^{3} + a\right )}^{\frac {5}{2}} x^{5}} \,d x } \] Input:
integrate((B*x^3+A)/x^5/(b*x^3+a)^(5/2),x, algorithm="giac")
Output:
integrate((B*x^3 + A)/((b*x^3 + a)^(5/2)*x^5), x)
Timed out. \[ \int \frac {A+B x^3}{x^5 \left (a+b x^3\right )^{5/2}} \, dx=\int \frac {B\,x^3+A}{x^5\,{\left (b\,x^3+a\right )}^{5/2}} \,d x \] Input:
int((A + B*x^3)/(x^5*(a + b*x^3)^(5/2)),x)
Output:
int((A + B*x^3)/(x^5*(a + b*x^3)^(5/2)), x)
\[ \int \frac {A+B x^3}{x^5 \left (a+b x^3\right )^{5/2}} \, dx=\int \frac {\sqrt {b \,x^{3}+a}}{b^{2} x^{11}+2 a b \,x^{8}+a^{2} x^{5}}d x \] Input:
int((B*x^3+A)/x^5/(b*x^3+a)^(5/2),x)
Output:
int(sqrt(a + b*x**3)/(a**2*x**5 + 2*a*b*x**8 + b**2*x**11),x)