3.287 \(\int \frac {(\frac {e (a+b x^2)}{c+d x^2})^{3/2}}{x^4} \, dx\)

Optimal. Leaf size=383 \[ \frac {b e (3 b c-4 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a c^{3/2} \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {d} e (7 b c-8 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 c^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {e \left (c+d x^2\right ) (7 b c-8 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{3 c^3 x}+\frac {d e x (7 b c-8 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{3 c^3}+\frac {e \left (c+d x^2\right ) (3 b c-4 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{3 c^2 d x^3}-\frac {e (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x^3} \]

[Out]

-(-a*d+b*c)*e*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/c/d/x^3+1/3*d*(-8*a*d+7*b*c)*e*x*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/c^3
+1/3*(-4*a*d+3*b*c)*e*(d*x^2+c)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/c^2/d/x^3-1/3*(-8*a*d+7*b*c)*e*(d*x^2+c)*(e*(b*x
^2+a)/(d*x^2+c))^(1/2)/c^3/x+1/3*b*(-4*a*d+3*b*c)*e*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*EllipticF(x*d^(1/2
)/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/a/c^(3/2)/d^(1/2)/(c*(b*x^2+a)/a/
(d*x^2+c))^(1/2)-1/3*(-8*a*d+7*b*c)*e*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*EllipticE(x*d^(1/2)/c^(1/2)/(1+d
*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))*d^(1/2)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/c^(5/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)

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Rubi [A]  time = 0.63, antiderivative size = 383, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {6719, 468, 583, 531, 418, 492, 411} \[ -\frac {e \left (c+d x^2\right ) (7 b c-8 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{3 c^3 x}+\frac {e \left (c+d x^2\right ) (3 b c-4 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{3 c^2 d x^3}+\frac {d e x (7 b c-8 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{3 c^3}+\frac {b e (3 b c-4 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a c^{3/2} \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {d} e (7 b c-8 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 c^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {e (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x^3} \]

Antiderivative was successfully verified.

[In]

Int[((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^4,x]

[Out]

-(((b*c - a*d)*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(c*d*x^3)) + (d*(7*b*c - 8*a*d)*e*x*Sqrt[(e*(a + b*x^2))/(
c + d*x^2)])/(3*c^3) + ((3*b*c - 4*a*d)*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(c + d*x^2))/(3*c^2*d*x^3) - ((7*b
*c - 8*a*d)*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(c + d*x^2))/(3*c^3*x) - (Sqrt[d]*(7*b*c - 8*a*d)*e*Sqrt[(e*(a
+ b*x^2))/(c + d*x^2)]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*c^(5/2)*Sqrt[(c*(a + b*x^2
))/(a*(c + d*x^2))]) + (b*(3*b*c - 4*a*d)*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqr
t[c]], 1 - (b*c)/(a*d)])/(3*a*c^(3/2)*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))])

Rule 411

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticE[ArcTan
[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticF[ArcT
an[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 468

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[((c*b -
a*d)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1))/(a*b*e*n*(p + 1)), x] + Dist[1/(a*b*n*(p + 1)), I
nt[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c*b - a*d)*(m + 1)) + d*(c*b*n*(p
+ 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 492

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(x*Sqrt[a + b*x^2])/(b*Sqr
t[c + d*x^2]), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rule 531

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Dist[
e, Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x] + Dist[f, Int[x^n*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, n, p, q}, x]

Rule 583

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rubi steps

\begin {align*} \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^4} \, dx &=\frac {\left (e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {\left (a+b x^2\right )^{3/2}}{x^4 \left (c+d x^2\right )^{3/2}} \, dx}{\sqrt {a+b x^2}}\\ &=-\frac {(b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x^3}-\frac {\left (e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {a (3 b c-4 a d)+b (2 b c-3 a d) x^2}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{c d \sqrt {a+b x^2}}\\ &=-\frac {(b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x^3}+\frac {(3 b c-4 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 c^2 d x^3}+\frac {\left (e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {a^2 d (7 b c-8 a d)+a b d (3 b c-4 a d) x^2}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 a c^2 d \sqrt {a+b x^2}}\\ &=-\frac {(b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x^3}+\frac {(3 b c-4 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 c^2 d x^3}-\frac {(7 b c-8 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 c^3 x}-\frac {\left (e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {-a^2 b c d (3 b c-4 a d)-a^2 b d^2 (7 b c-8 a d) x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 a^2 c^3 d \sqrt {a+b x^2}}\\ &=-\frac {(b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x^3}+\frac {(3 b c-4 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 c^2 d x^3}-\frac {(7 b c-8 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 c^3 x}+\frac {\left (b d (7 b c-8 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 c^3 \sqrt {a+b x^2}}+\frac {\left (b (3 b c-4 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 c^2 \sqrt {a+b x^2}}\\ &=-\frac {(b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x^3}+\frac {d (7 b c-8 a d) e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{3 c^3}+\frac {(3 b c-4 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 c^2 d x^3}-\frac {(7 b c-8 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 c^3 x}+\frac {b (3 b c-4 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a c^{3/2} \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\left (d (7 b c-8 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}\right ) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c^2 \sqrt {a+b x^2}}\\ &=-\frac {(b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x^3}+\frac {d (7 b c-8 a d) e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{3 c^3}+\frac {(3 b c-4 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 c^2 d x^3}-\frac {(7 b c-8 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 c^3 x}-\frac {\sqrt {d} (7 b c-8 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 c^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {b (3 b c-4 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a c^{3/2} \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\\ \end {align*}

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Mathematica [C]  time = 0.47, size = 275, normalized size = 0.72 \[ \frac {e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\sqrt {\frac {b}{a}} \left (a^2 \left (c^2-4 c d x^2-8 d^2 x^4\right )+a b x^2 \left (5 c^2+3 c d x^2-8 d^2 x^4\right )+b^2 c x^4 \left (4 c+7 d x^2\right )\right )-4 i b c x^3 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} (a d-b c) F\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i b c x^3 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} (8 a d-7 b c) E\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )\right )}{3 c^3 x^3 \sqrt {\frac {b}{a}} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^4,x]

[Out]

(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(-(Sqrt[b/a]*(b^2*c*x^4*(4*c + 7*d*x^2) + a^2*(c^2 - 4*c*d*x^2 - 8*d^2*x^
4) + a*b*x^2*(5*c^2 + 3*c*d*x^2 - 8*d^2*x^4))) + I*b*c*(-7*b*c + 8*a*d)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^
2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (4*I)*b*c*(-(b*c) + a*d)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1
+ (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(3*Sqrt[b/a]*c^3*x^3*(a + b*x^2))

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fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b e x^{2} + a e\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{d x^{6} + c x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*(b*x^2+a)/(d*x^2+c))^(3/2)/x^4,x, algorithm="fricas")

[Out]

integral((b*e*x^2 + a*e)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c))/(d*x^6 + c*x^4), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*(b*x^2+a)/(d*x^2+c))^(3/2)/x^4,x, algorithm="giac")

[Out]

integrate(((b*x^2 + a)*e/(d*x^2 + c))^(3/2)/x^4, x)

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maple [A]  time = 0.04, size = 791, normalized size = 2.07 \[ \frac {\left (\frac {\left (b \,x^{2}+a \right ) e}{d \,x^{2}+c}\right )^{\frac {3}{2}} \left (d \,x^{2}+c \right ) \left (5 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{6}+3 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{6}-4 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, b^{2} c d \,x^{6}-3 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, b^{2} c d \,x^{6}+5 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a^{2} d^{2} x^{4}+3 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a^{2} d^{2} x^{4}-3 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a b c d \,x^{4}-8 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a b c d \,x^{3} \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+4 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a b c d \,x^{3} \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-4 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, b^{2} c^{2} x^{4}+7 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, b^{2} c^{2} x^{3} \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-4 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, b^{2} c^{2} x^{3} \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+4 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a^{2} c d \,x^{2}-5 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a b \,c^{2} x^{2}-\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a^{2} c^{2}\right )}{3 \left (b \,x^{2}+a \right )^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, c^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x^2+a)/(d*x^2+c)*e)^(3/2)/x^4,x)

[Out]

1/3*((b*x^2+a)/(d*x^2+c)*e)^(3/2)*(d*x^2+c)*(5*((d*x^2+c)*(b*x^2+a))^(1/2)*(-1/a*b)^(1/2)*x^6*a*b*d^2-4*((d*x^
2+c)*(b*x^2+a))^(1/2)*(-1/a*b)^(1/2)*x^6*b^2*c*d+3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(-1/a*b)^(1/2)*x^6*a*b*
d^2-3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(-1/a*b)^(1/2)*x^6*b^2*c*d+4*((d*x^2+c)*(b*x^2+a))^(1/2)*((b*x^2+a)/
a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF((-1/a*b)^(1/2)*x,(a/b/c*d)^(1/2))*x^3*a*b*c*d-4*((d*x^2+c)*(b*x^2+a))^(
1/2)*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF((-1/a*b)^(1/2)*x,(a/b/c*d)^(1/2))*x^3*b^2*c^2-8*((d*x^2
+c)*(b*x^2+a))^(1/2)*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE((-1/a*b)^(1/2)*x,(a/b/c*d)^(1/2))*x^3*a
*b*c*d+7*((d*x^2+c)*(b*x^2+a))^(1/2)*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE((-1/a*b)^(1/2)*x,(a/b/c
*d)^(1/2))*x^3*b^2*c^2+5*((d*x^2+c)*(b*x^2+a))^(1/2)*(-1/a*b)^(1/2)*x^4*a^2*d^2-4*((d*x^2+c)*(b*x^2+a))^(1/2)*
(-1/a*b)^(1/2)*x^4*b^2*c^2+3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(-1/a*b)^(1/2)*x^4*a^2*d^2-3*(b*d*x^4+a*d*x^2
+b*c*x^2+a*c)^(1/2)*(-1/a*b)^(1/2)*x^4*a*b*c*d+4*((d*x^2+c)*(b*x^2+a))^(1/2)*(-1/a*b)^(1/2)*x^2*a^2*c*d-5*((d*
x^2+c)*(b*x^2+a))^(1/2)*(-1/a*b)^(1/2)*x^2*a*b*c^2-((d*x^2+c)*(b*x^2+a))^(1/2)*(-1/a*b)^(1/2)*a^2*c^2)/(b*x^2+
a)^2/c^3/x^3/(-1/a*b)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*(b*x^2+a)/(d*x^2+c))^(3/2)/x^4,x, algorithm="maxima")

[Out]

integrate(((b*x^2 + a)*e/(d*x^2 + c))^(3/2)/x^4, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}\right )}^{3/2}}{x^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^4,x)

[Out]

int(((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^4, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*(b*x**2+a)/(d*x**2+c))**(3/2)/x**4,x)

[Out]

Timed out

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