Integrand size = 19, antiderivative size = 362 \[ \int \frac {1}{(c x)^{7/2} \left (a+b x^2\right )^{5/2}} \, dx=\frac {1}{3 a c (c x)^{5/2} \left (a+b x^2\right )^{3/2}}+\frac {11}{6 a^2 c (c x)^{5/2} \sqrt {a+b x^2}}-\frac {77 \sqrt {a+b x^2}}{30 a^3 c (c x)^{5/2}}+\frac {77 b \sqrt {a+b x^2}}{10 a^4 c^3 \sqrt {c x}}-\frac {77 b^{3/2} \sqrt {c x} \sqrt {a+b x^2}}{10 a^4 c^4 \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {77 b^{5/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{10 a^{15/4} c^{7/2} \sqrt {a+b x^2}}-\frac {77 b^{5/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{20 a^{15/4} c^{7/2} \sqrt {a+b x^2}} \] Output:
1/3/a/c/(c*x)^(5/2)/(b*x^2+a)^(3/2)+11/6/a^2/c/(c*x)^(5/2)/(b*x^2+a)^(1/2) -77/30*(b*x^2+a)^(1/2)/a^3/c/(c*x)^(5/2)+77/10*b*(b*x^2+a)^(1/2)/a^4/c^3/( c*x)^(1/2)-77/10*b^(3/2)*(c*x)^(1/2)*(b*x^2+a)^(1/2)/a^4/c^4/(a^(1/2)+b^(1 /2)*x)+77/10*b^(5/4)*(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2) ^(1/2)*EllipticE(sin(2*arctan(b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2))),1/2*2^ (1/2))/a^(15/4)/c^(7/2)/(b*x^2+a)^(1/2)-77/20*b^(5/4)*(a^(1/2)+b^(1/2)*x)* ((b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2)*InverseJacobiAM(2*arctan(b^(1/4)*( c*x)^(1/2)/a^(1/4)/c^(1/2)),1/2*2^(1/2))/a^(15/4)/c^(7/2)/(b*x^2+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.16 \[ \int \frac {1}{(c x)^{7/2} \left (a+b x^2\right )^{5/2}} \, dx=-\frac {2 x \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {5}{2},-\frac {1}{4},-\frac {b x^2}{a}\right )}{5 a^2 (c x)^{7/2} \sqrt {a+b x^2}} \] Input:
Integrate[1/((c*x)^(7/2)*(a + b*x^2)^(5/2)),x]
Output:
(-2*x*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[-5/4, 5/2, -1/4, -((b*x^2)/a)] )/(5*a^2*(c*x)^(7/2)*Sqrt[a + b*x^2])
Time = 0.42 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.15, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {253, 253, 264, 264, 266, 834, 27, 761, 1510}
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.
| \(\displaystyle \int \frac {1}{(c x)^{7/2} \left (a+b x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {11 \int \frac {1}{(c x)^{7/2} \left (b x^2+a\right )^{3/2}}dx}{6 a}+\frac {1}{3 a c (c x)^{5/2} \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {11 \left (\frac {7 \int \frac {1}{(c x)^{7/2} \sqrt {b x^2+a}}dx}{2 a}+\frac {1}{a c (c x)^{5/2} \sqrt {a+b x^2}}\right )}{6 a}+\frac {1}{3 a c (c x)^{5/2} \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (-\frac {3 b \int \frac {1}{(c x)^{3/2} \sqrt {b x^2+a}}dx}{5 a c^2}-\frac {2 \sqrt {a+b x^2}}{5 a c (c x)^{5/2}}\right )}{2 a}+\frac {1}{a c (c x)^{5/2} \sqrt {a+b x^2}}\right )}{6 a}+\frac {1}{3 a c (c x)^{5/2} \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (-\frac {3 b \left (\frac {b \int \frac {\sqrt {c x}}{\sqrt {b x^2+a}}dx}{a c^2}-\frac {2 \sqrt {a+b x^2}}{a c \sqrt {c x}}\right )}{5 a c^2}-\frac {2 \sqrt {a+b x^2}}{5 a c (c x)^{5/2}}\right )}{2 a}+\frac {1}{a c (c x)^{5/2} \sqrt {a+b x^2}}\right )}{6 a}+\frac {1}{3 a c (c x)^{5/2} \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (-\frac {3 b \left (\frac {2 b \int \frac {c x}{\sqrt {b x^2+a}}d\sqrt {c x}}{a c^3}-\frac {2 \sqrt {a+b x^2}}{a c \sqrt {c x}}\right )}{5 a c^2}-\frac {2 \sqrt {a+b x^2}}{5 a c (c x)^{5/2}}\right )}{2 a}+\frac {1}{a c (c x)^{5/2} \sqrt {a+b x^2}}\right )}{6 a}+\frac {1}{3 a c (c x)^{5/2} \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (-\frac {3 b \left (\frac {2 b \left (\frac {\sqrt {a} c \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {c x}}{\sqrt {b}}-\frac {\sqrt {a} c \int \frac {\sqrt {a} c-\sqrt {b} c x}{\sqrt {a} c \sqrt {b x^2+a}}d\sqrt {c x}}{\sqrt {b}}\right )}{a c^3}-\frac {2 \sqrt {a+b x^2}}{a c \sqrt {c x}}\right )}{5 a c^2}-\frac {2 \sqrt {a+b x^2}}{5 a c (c x)^{5/2}}\right )}{2 a}+\frac {1}{a c (c x)^{5/2} \sqrt {a+b x^2}}\right )}{6 a}+\frac {1}{3 a c (c x)^{5/2} \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (-\frac {3 b \left (\frac {2 b \left (\frac {\sqrt {a} c \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {c x}}{\sqrt {b}}-\frac {\int \frac {\sqrt {a} c-\sqrt {b} c x}{\sqrt {b x^2+a}}d\sqrt {c x}}{\sqrt {b}}\right )}{a c^3}-\frac {2 \sqrt {a+b x^2}}{a c \sqrt {c x}}\right )}{5 a c^2}-\frac {2 \sqrt {a+b x^2}}{5 a c (c x)^{5/2}}\right )}{2 a}+\frac {1}{a c (c x)^{5/2} \sqrt {a+b x^2}}\right )}{6 a}+\frac {1}{3 a c (c x)^{5/2} \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (-\frac {3 b \left (\frac {2 b \left (\frac {\sqrt [4]{a} \sqrt {c} \left (\sqrt {a} c+\sqrt {b} c x\right ) \sqrt {\frac {a c^2+b c^2 x^2}{\left (\sqrt {a} c+\sqrt {b} c x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^2}}-\frac {\int \frac {\sqrt {a} c-\sqrt {b} c x}{\sqrt {b x^2+a}}d\sqrt {c x}}{\sqrt {b}}\right )}{a c^3}-\frac {2 \sqrt {a+b x^2}}{a c \sqrt {c x}}\right )}{5 a c^2}-\frac {2 \sqrt {a+b x^2}}{5 a c (c x)^{5/2}}\right )}{2 a}+\frac {1}{a c (c x)^{5/2} \sqrt {a+b x^2}}\right )}{6 a}+\frac {1}{3 a c (c x)^{5/2} \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (-\frac {3 b \left (\frac {2 b \left (\frac {\sqrt [4]{a} \sqrt {c} \left (\sqrt {a} c+\sqrt {b} c x\right ) \sqrt {\frac {a c^2+b c^2 x^2}{\left (\sqrt {a} c+\sqrt {b} c x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^2}}-\frac {\frac {\sqrt [4]{a} \sqrt {c} \left (\sqrt {a} c+\sqrt {b} c x\right ) \sqrt {\frac {a c^2+b c^2 x^2}{\left (\sqrt {a} c+\sqrt {b} c x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {c^2 \sqrt {c x} \sqrt {a+b x^2}}{\sqrt {a} c+\sqrt {b} c x}}{\sqrt {b}}\right )}{a c^3}-\frac {2 \sqrt {a+b x^2}}{a c \sqrt {c x}}\right )}{5 a c^2}-\frac {2 \sqrt {a+b x^2}}{5 a c (c x)^{5/2}}\right )}{2 a}+\frac {1}{a c (c x)^{5/2} \sqrt {a+b x^2}}\right )}{6 a}+\frac {1}{3 a c (c x)^{5/2} \left (a+b x^2\right )^{3/2}}\) |
Input:
Int[1/((c*x)^(7/2)*(a + b*x^2)^(5/2)),x]
Output:
1/(3*a*c*(c*x)^(5/2)*(a + b*x^2)^(3/2)) + (11*(1/(a*c*(c*x)^(5/2)*Sqrt[a + b*x^2]) + (7*((-2*Sqrt[a + b*x^2])/(5*a*c*(c*x)^(5/2)) - (3*b*((-2*Sqrt[a + b*x^2])/(a*c*Sqrt[c*x]) + (2*b*(-((-((c^2*Sqrt[c*x]*Sqrt[a + b*x^2])/(S qrt[a]*c + Sqrt[b]*c*x)) + (a^(1/4)*Sqrt[c]*(Sqrt[a]*c + Sqrt[b]*c*x)*Sqrt [(a*c^2 + b*c^2*x^2)/(Sqrt[a]*c + Sqrt[b]*c*x)^2]*EllipticE[2*ArcTan[(b^(1 /4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])], 1/2])/(b^(1/4)*Sqrt[a + b*x^2]))/Sqrt[b ]) + (a^(1/4)*Sqrt[c]*(Sqrt[a]*c + Sqrt[b]*c*x)*Sqrt[(a*c^2 + b*c^2*x^2)/( Sqrt[a]*c + Sqrt[b]*c*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4 )*Sqrt[c])], 1/2])/(2*b^(3/4)*Sqrt[a + b*x^2])))/(a*c^3)))/(5*a*c^2)))/(2* a)))/(6*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S imp[1/q Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Time = 4.68 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.86
| method | result | size |
| elliptic | \(\frac {\sqrt {c x \left (b \,x^{2}+a \right )}\, \left (-\frac {2 \sqrt {b c \,x^{3}+a c x}}{5 a^{3} c^{4} x^{3}}+\frac {26 \left (x^{2} b c +a c \right ) b}{5 a^{4} c^{4} \sqrt {x \left (x^{2} b c +a c \right )}}+\frac {x \sqrt {b c \,x^{3}+a c x}}{3 a^{3} c^{4} \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {5 b^{2} x^{2}}{2 c^{3} a^{4} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b c x}}-\frac {77 b \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{20 a^{4} c^{3} \sqrt {b c \,x^{3}+a c x}}\right )}{\sqrt {c x}\, \sqrt {b \,x^{2}+a}}\) | \(312\) |
| default | \(-\frac {462 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a \,b^{2} x^{4}-231 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a \,b^{2} x^{4}+462 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b \,x^{2}-231 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b \,x^{2}-462 b^{3} x^{6}-770 a \,b^{2} x^{4}-264 a^{2} b \,x^{2}+24 a^{3}}{60 x^{2} a^{4} c^{3} \sqrt {c x}\, \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(410\) |
| risch | \(-\frac {2 \sqrt {b \,x^{2}+a}\, \left (-13 b \,x^{2}+a \right )}{5 a^{4} x^{2} c^{3} \sqrt {c x}}-\frac {b^{2} \left (\frac {13 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {b c \,x^{3}+a c x}}-10 a \left (\frac {x^{2}}{a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b c x}}-\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{2 a b \sqrt {b c \,x^{3}+a c x}}\right )-5 a^{2} \left (\frac {x \sqrt {b c \,x^{3}+a c x}}{3 a c \,b^{2} \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {x^{2}}{2 a^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b c x}}-\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{4 a^{2} b \sqrt {b c \,x^{3}+a c x}}\right )\right ) \sqrt {c x \left (b \,x^{2}+a \right )}}{5 a^{4} c^{3} \sqrt {c x}\, \sqrt {b \,x^{2}+a}}\) | \(650\) |
Input:
int(1/(c*x)^(7/2)/(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
(c*x*(b*x^2+a))^(1/2)/(c*x)^(1/2)/(b*x^2+a)^(1/2)*(-2/5/a^3/c^4*(b*c*x^3+a *c*x)^(1/2)/x^3+26/5*(b*c*x^2+a*c)*b/a^4/c^4/(x*(b*c*x^2+a*c))^(1/2)+1/3/a ^3/c^4*x*(b*c*x^3+a*c*x)^(1/2)/(x^2+a/b)^2+5/2*b^2/c^3*x^2/a^4/((x^2+a/b)* b*c*x)^(1/2)-77/20*b/a^4/c^3*(-a*b)^(1/2)*((x+1/b*(-a*b)^(1/2))*b/(-a*b)^( 1/2))^(1/2)*(-2*(x-1/b*(-a*b)^(1/2))*b/(-a*b)^(1/2))^(1/2)*(-b/(-a*b)^(1/2 )*x)^(1/2)/(b*c*x^3+a*c*x)^(1/2)*(-2/b*(-a*b)^(1/2)*EllipticE(((x+1/b*(-a* b)^(1/2))*b/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))+1/b*(-a*b)^(1/2)*EllipticF((( x+1/b*(-a*b)^(1/2))*b/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))))
Time = 0.09 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.38 \[ \int \frac {1}{(c x)^{7/2} \left (a+b x^2\right )^{5/2}} \, dx=\frac {231 \, {\left (b^{3} x^{7} + 2 \, a b^{2} x^{5} + a^{2} b x^{3}\right )} \sqrt {b c} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (231 \, b^{3} x^{6} + 385 \, a b^{2} x^{4} + 132 \, a^{2} b x^{2} - 12 \, a^{3}\right )} \sqrt {b x^{2} + a} \sqrt {c x}}{30 \, {\left (a^{4} b^{2} c^{4} x^{7} + 2 \, a^{5} b c^{4} x^{5} + a^{6} c^{4} x^{3}\right )}} \] Input:
integrate(1/(c*x)^(7/2)/(b*x^2+a)^(5/2),x, algorithm="fricas")
Output:
1/30*(231*(b^3*x^7 + 2*a*b^2*x^5 + a^2*b*x^3)*sqrt(b*c)*weierstrassZeta(-4 *a/b, 0, weierstrassPInverse(-4*a/b, 0, x)) + (231*b^3*x^6 + 385*a*b^2*x^4 + 132*a^2*b*x^2 - 12*a^3)*sqrt(b*x^2 + a)*sqrt(c*x))/(a^4*b^2*c^4*x^7 + 2 *a^5*b*c^4*x^5 + a^6*c^4*x^3)
Result contains complex when optimal does not.
Time = 24.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.14 \[ \int \frac {1}{(c x)^{7/2} \left (a+b x^2\right )^{5/2}} \, dx=\frac {\Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {5}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} c^{\frac {7}{2}} x^{\frac {5}{2}} \Gamma \left (- \frac {1}{4}\right )} \] Input:
integrate(1/(c*x)**(7/2)/(b*x**2+a)**(5/2),x)
Output:
gamma(-5/4)*hyper((-5/4, 5/2), (-1/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(5 /2)*c**(7/2)*x**(5/2)*gamma(-1/4))
\[ \int \frac {1}{(c x)^{7/2} \left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} \left (c x\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/(c*x)^(7/2)/(b*x^2+a)^(5/2),x, algorithm="maxima")
Output:
integrate(1/((b*x^2 + a)^(5/2)*(c*x)^(7/2)), x)
\[ \int \frac {1}{(c x)^{7/2} \left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} \left (c x\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/(c*x)^(7/2)/(b*x^2+a)^(5/2),x, algorithm="giac")
Output:
integrate(1/((b*x^2 + a)^(5/2)*(c*x)^(7/2)), x)
Timed out. \[ \int \frac {1}{(c x)^{7/2} \left (a+b x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (c\,x\right )}^{7/2}\,{\left (b\,x^2+a\right )}^{5/2}} \,d x \] Input:
int(1/((c*x)^(7/2)*(a + b*x^2)^(5/2)),x)
Output:
int(1/((c*x)^(7/2)*(a + b*x^2)^(5/2)), x)
\[ \int \frac {1}{(c x)^{7/2} \left (a+b x^2\right )^{5/2}} \, dx=\frac {\sqrt {c}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{10}+3 a \,b^{2} x^{8}+3 a^{2} b \,x^{6}+a^{3} x^{4}}d x \right )}{c^{4}} \] Input:
int(1/(c*x)^(7/2)/(b*x^2+a)^(5/2),x)
Output:
(sqrt(c)*int((sqrt(x)*sqrt(a + b*x**2))/(a**3*x**4 + 3*a**2*b*x**6 + 3*a*b **2*x**8 + b**3*x**10),x))/c**4