Integrand size = 30, antiderivative size = 330 \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{5/2} \left (a-b x^2\right )} \, dx=-\frac {2 c \sqrt {c-d x^2}}{3 a e (e x)^{3/2}}+\frac {2 \sqrt [4]{c} d^{3/4} (b c-3 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{3 a b e^{5/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d)^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^2 b \sqrt [4]{d} e^{5/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d)^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^2 b \sqrt [4]{d} e^{5/2} \sqrt {c-d x^2}} \] Output:
-2/3*c*(-d*x^2+c)^(1/2)/a/e/(e*x)^(3/2)+2/3*c^(1/4)*d^(3/4)*(-3*a*d+b*c)*( 1-d*x^2/c)^(1/2)*EllipticF(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)/a/b/e^(5 /2)/(-d*x^2+c)^(1/2)+c^(1/4)*(-a*d+b*c)^2*(1-d*x^2/c)^(1/2)*EllipticPi(d^( 1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),-b^(1/2)*c^(1/2)/a^(1/2)/d^(1/2),I)/a^2/b /d^(1/4)/e^(5/2)/(-d*x^2+c)^(1/2)+c^(1/4)*(-a*d+b*c)^2*(1-d*x^2/c)^(1/2)*E llipticPi(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),b^(1/2)*c^(1/2)/a^(1/2)/d^(1 /2),I)/a^2/b/d^(1/4)/e^(5/2)/(-d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.12 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.46 \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{5/2} \left (a-b x^2\right )} \, dx=\frac {x \left (-10 a c \left (c-d x^2\right )+10 c (3 b c-5 a d) x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )-2 d (b c-3 a d) x^4 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{15 a^2 (e x)^{5/2} \sqrt {c-d x^2}} \] Input:
Integrate[(c - d*x^2)^(3/2)/((e*x)^(5/2)*(a - b*x^2)),x]
Output:
(x*(-10*a*c*(c - d*x^2) + 10*c*(3*b*c - 5*a*d)*x^2*Sqrt[1 - (d*x^2)/c]*App ellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] - 2*d*(b*c - 3*a*d)*x^4*Sqrt [1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(15*a^2 *(e*x)^(5/2)*Sqrt[c - d*x^2])
Time = 0.71 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {368, 27, 974, 27, 1021, 765, 762, 925, 27, 1543, 1542}
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.
| \(\displaystyle \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{5/2} \left (a-b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \int \frac {\left (c-d x^2\right )^{3/2}}{x^2 \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e \int \frac {\left (c-d x^2\right )^{3/2}}{e^2 x^2 \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}\) |
| \(\Big \downarrow \) 974 |
| \(\displaystyle 2 e \left (\frac {\int \frac {c (3 b c-5 a d) e^2-d (b c-3 a d) e^2 x^2}{e^2 \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 a e^2}-\frac {c \sqrt {c-d x^2}}{3 a e^2 (e x)^{3/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e \left (\frac {\int \frac {c (3 b c-5 a d) e^2-d (b c-3 a d) e^2 x^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 a e^4}-\frac {c \sqrt {c-d x^2}}{3 a e^2 (e x)^{3/2}}\right )\) |
| \(\Big \downarrow \) 1021 |
| \(\displaystyle 2 e \left (\frac {\frac {3 e^2 (b c-a d)^2 \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}+\frac {d (b c-3 a d) \int \frac {1}{\sqrt {c-d x^2}}d\sqrt {e x}}{b}}{3 a e^4}-\frac {c \sqrt {c-d x^2}}{3 a e^2 (e x)^{3/2}}\right )\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle 2 e \left (\frac {\frac {3 e^2 (b c-a d)^2 \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}+\frac {d \sqrt {1-\frac {d x^2}{c}} (b c-3 a d) \int \frac {1}{\sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{b \sqrt {c-d x^2}}}{3 a e^4}-\frac {c \sqrt {c-d x^2}}{3 a e^2 (e x)^{3/2}}\right )\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle 2 e \left (\frac {\frac {3 e^2 (b c-a d)^2 \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (b c-3 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt {c-d x^2}}}{3 a e^4}-\frac {c \sqrt {c-d x^2}}{3 a e^2 (e x)^{3/2}}\right )\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle 2 e \left (\frac {\frac {3 e^2 (b c-a d)^2 \left (\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}+\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}\right )}{b}+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (b c-3 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt {c-d x^2}}}{3 a e^4}-\frac {c \sqrt {c-d x^2}}{3 a e^2 (e x)^{3/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e \left (\frac {\frac {3 e^2 (b c-a d)^2 \left (\frac {\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}+\frac {\int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}\right )}{b}+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (b c-3 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt {c-d x^2}}}{3 a e^4}-\frac {c \sqrt {c-d x^2}}{3 a e^2 (e x)^{3/2}}\right )\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle 2 e \left (\frac {\frac {3 e^2 (b c-a d)^2 \left (\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}+\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}\right )}{b}+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (b c-3 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt {c-d x^2}}}{3 a e^4}-\frac {c \sqrt {c-d x^2}}{3 a e^2 (e x)^{3/2}}\right )\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle 2 e \left (\frac {\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (b c-3 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt {c-d x^2}}+\frac {3 e^2 (b c-a d)^2 \left (\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}\right )}{b}}{3 a e^4}-\frac {c \sqrt {c-d x^2}}{3 a e^2 (e x)^{3/2}}\right )\) |
Input:
Int[(c - d*x^2)^(3/2)/((e*x)^(5/2)*(a - b*x^2)),x]
Output:
2*e*(-1/3*(c*Sqrt[c - d*x^2])/(a*e^2*(e*x)^(3/2)) + ((c^(1/4)*d^(3/4)*(b*c - 3*a*d)*Sqrt[e]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x]) /(c^(1/4)*Sqrt[e])], -1])/(b*Sqrt[c - d*x^2]) + (3*(b*c - a*d)^2*e^2*((c^( 1/4)*Sqrt[1 - (d*x^2)/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])) , ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*a*d^(1/4)*e^(3/2) *Sqrt[c - d*x^2]) + (c^(1/4)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*Sqrt[ c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1]) /(2*a*d^(1/4)*e^(3/2)*Sqrt[c - d*x^2])))/b)/(3*a*e^4))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*2)/e^2))^p*(c + d*(x^(k*2)/e^2))^q, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[m ] && IntegerQ[p]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt [a + b*x^4] Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ [b/a] && !GtQ[a, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Simp[ 1/(2*c) Int[1/(Sqrt[a + b*x^4]*(1 - Rt[-d/c, 2]*x^2)), x], x] + Simp[1/(2 *c) Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) )^(q_), x_Symbol] :> Simp[c*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^ (q - 1)/(a*e*(m + 1))), x] - Simp[1/(a*e^n*(m + 1)) Int[(e*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*(c*b - a*d)*(m + 1) + c*n*(b*c*(p + 1 ) + a*d*(q - 1)) + d*((c*b - a*d)*(m + 1) + c*b*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q , 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x _)^(n_)]), x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^n], x], x] + Simp[(b* e - a*f)/b Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x ], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a) ]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && !GtQ[a, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1143\) vs. \(2(252)=504\).
Time = 1.10 (sec) , antiderivative size = 1144, normalized size of antiderivative = 3.47
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1144\) |
| default | \(\text {Expression too large to display}\) | \(1729\) |
Input:
int((-d*x^2+c)^(3/2)/(e*x)^(5/2)/(-b*x^2+a),x,method=_RETURNVERBOSE)
Output:
((-d*x^2+c)*e*x)^(1/2)/(e*x)^(1/2)/(-d*x^2+c)^(1/2)*(-2/3/e^3*c/a*(-d*e*x^ 3+c*e*x)^(1/2)/x^2-d*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^ (1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)*EllipticF(((x +1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))/e^2/b+1/3*(c*d)^(1/2)* (1+x*d/(c*d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^( 1/2)/(-d*e*x^3+c*e*x)^(1/2)*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^ (1/2),1/2*2^(1/2))*c/e^2/a-1/2*a/b/e^2/(a*b)^(1/2)*d*(c*d)^(1/2)*(1+x*d/(c *d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)/(-d* e*x^3+c*e*x)^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2))*EllipticPi(((x+1/d*( c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a *b)^(1/2)),1/2*2^(1/2))+1/e^2/(a*b)^(1/2)*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^ (1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x )^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2))*EllipticPi(((x+1/d*(c*d)^(1/2)) *d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2)), 1/2*2^(1/2))*c-1/2/a*b/e^2/(a*b)^(1/2)/d*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^( 1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x) ^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2))*EllipticPi(((x+1/d*(c*d)^(1/2))* d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2)),1 /2*2^(1/2))*c^2+1/2*a/b/e^2/(a*b)^(1/2)*d*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^ (1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*...
Timed out. \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{5/2} \left (a-b x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((-d*x^2+c)^(3/2)/(e*x)^(5/2)/(-b*x^2+a),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{5/2} \left (a-b x^2\right )} \, dx=- \int \frac {c \sqrt {c - d x^{2}}}{- a \left (e x\right )^{\frac {5}{2}} + b x^{2} \left (e x\right )^{\frac {5}{2}}}\, dx - \int \left (- \frac {d x^{2} \sqrt {c - d x^{2}}}{- a \left (e x\right )^{\frac {5}{2}} + b x^{2} \left (e x\right )^{\frac {5}{2}}}\right )\, dx \] Input:
integrate((-d*x**2+c)**(3/2)/(e*x)**(5/2)/(-b*x**2+a),x)
Output:
-Integral(c*sqrt(c - d*x**2)/(-a*(e*x)**(5/2) + b*x**2*(e*x)**(5/2)), x) - Integral(-d*x**2*sqrt(c - d*x**2)/(-a*(e*x)**(5/2) + b*x**2*(e*x)**(5/2)) , x)
\[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{5/2} \left (a-b x^2\right )} \, dx=\int { -\frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} - a\right )} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((-d*x^2+c)^(3/2)/(e*x)^(5/2)/(-b*x^2+a),x, algorithm="maxima")
Output:
-integrate((-d*x^2 + c)^(3/2)/((b*x^2 - a)*(e*x)^(5/2)), x)
\[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{5/2} \left (a-b x^2\right )} \, dx=\int { -\frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} - a\right )} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((-d*x^2+c)^(3/2)/(e*x)^(5/2)/(-b*x^2+a),x, algorithm="giac")
Output:
integrate(-(-d*x^2 + c)^(3/2)/((b*x^2 - a)*(e*x)^(5/2)), x)
Timed out. \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{5/2} \left (a-b x^2\right )} \, dx=\int \frac {{\left (c-d\,x^2\right )}^{3/2}}{{\left (e\,x\right )}^{5/2}\,\left (a-b\,x^2\right )} \,d x \] Input:
int((c - d*x^2)^(3/2)/((e*x)^(5/2)*(a - b*x^2)),x)
Output:
int((c - d*x^2)^(3/2)/((e*x)^(5/2)*(a - b*x^2)), x)
\[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{5/2} \left (a-b x^2\right )} \, dx =\text {Too large to display} \] Input:
int((-d*x^2+c)^(3/2)/(e*x)^(5/2)/(-b*x^2+a),x)
Output:
(sqrt(e)*( - 2*sqrt(c - d*x**2)*a*c*d - 6*sqrt(c - d*x**2)*b*c**2 - 5*sqrt (x)*int(sqrt(c - d*x**2)/(sqrt(x)*a**2*c*d - sqrt(x)*a**2*d**2*x**2 + 3*sq rt(x)*a*b*c**2 - 4*sqrt(x)*a*b*c*d*x**2 + sqrt(x)*a*b*d**2*x**4 - 3*sqrt(x )*b**2*c**2*x**2 + 3*sqrt(x)*b**2*c*d*x**4),x)*a**3*c*d**3*x - 27*sqrt(x)* int(sqrt(c - d*x**2)/(sqrt(x)*a**2*c*d - sqrt(x)*a**2*d**2*x**2 + 3*sqrt(x )*a*b*c**2 - 4*sqrt(x)*a*b*c*d*x**2 + sqrt(x)*a*b*d**2*x**4 - 3*sqrt(x)*b* *2*c**2*x**2 + 3*sqrt(x)*b**2*c*d*x**4),x)*a**2*b*c**2*d**2*x - 27*sqrt(x) *int(sqrt(c - d*x**2)/(sqrt(x)*a**2*c*d - sqrt(x)*a**2*d**2*x**2 + 3*sqrt( x)*a*b*c**2 - 4*sqrt(x)*a*b*c*d*x**2 + sqrt(x)*a*b*d**2*x**4 - 3*sqrt(x)*b **2*c**2*x**2 + 3*sqrt(x)*b**2*c*d*x**4),x)*a*b**2*c**3*d*x + 27*sqrt(x)*i nt(sqrt(c - d*x**2)/(sqrt(x)*a**2*c*d - sqrt(x)*a**2*d**2*x**2 + 3*sqrt(x) *a*b*c**2 - 4*sqrt(x)*a*b*c*d*x**2 + sqrt(x)*a*b*d**2*x**4 - 3*sqrt(x)*b** 2*c**2*x**2 + 3*sqrt(x)*b**2*c*d*x**4),x)*b**3*c**4*x + 5*sqrt(x)*int((sqr t(c - d*x**2)*x**2)/(sqrt(x)*a**2*c*d - sqrt(x)*a**2*d**2*x**2 + 3*sqrt(x) *a*b*c**2 - 4*sqrt(x)*a*b*c*d*x**2 + sqrt(x)*a*b*d**2*x**4 - 3*sqrt(x)*b** 2*c**2*x**2 + 3*sqrt(x)*b**2*c*d*x**4),x)*a**2*b*c*d**3*x + 12*sqrt(x)*int ((sqrt(c - d*x**2)*x**2)/(sqrt(x)*a**2*c*d - sqrt(x)*a**2*d**2*x**2 + 3*sq rt(x)*a*b*c**2 - 4*sqrt(x)*a*b*c*d*x**2 + sqrt(x)*a*b*d**2*x**4 - 3*sqrt(x )*b**2*c**2*x**2 + 3*sqrt(x)*b**2*c*d*x**4),x)*a*b**2*c**2*d**2*x - 9*sqrt (x)*int((sqrt(c - d*x**2)*x**2)/(sqrt(x)*a**2*c*d - sqrt(x)*a**2*d**2*x...