3.269 \(\int \frac {\sqrt {\frac {e (a+b x^2)}{c+d x^2}}}{x^7} \, dx\)

Optimal. Leaf size=318 \[ -\frac {\left (-11 a^2 d^2+2 a b c d+b^2 c^2\right ) (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{16 a^2 c^3 \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {\sqrt {e} \left (5 a^2 d^2+2 a b c d+b^2 c^2\right ) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{16 a^{5/2} c^{7/2}}+\frac {(3 a d+b c) (b c-a d)^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 a c^3 \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )^2}+\frac {e^2 (b c-a d)^3 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{6 a c^2 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^3} \]

[Out]

1/6*(-a*d+b*c)^3*e^2*(e*(b*x^2+a)/(d*x^2+c))^(3/2)/a/c^2/(a*e-c*e*(b*x^2+a)/(d*x^2+c))^3-1/16*(-a*d+b*c)*(5*a^
2*d^2+2*a*b*c*d+b^2*c^2)*arctanh(c^(1/2)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/a^(1/2)/e^(1/2))*e^(1/2)/a^(5/2)/c^(7/2
)+1/8*(-a*d+b*c)^2*(3*a*d+b*c)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/a/c^3/(a-c*(b*x^2+a)/(d*x^2+c))^2-1/16*(-a*d+b*c)
*(-11*a^2*d^2+2*a*b*c*d+b^2*c^2)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/a^2/c^3/(a-c*(b*x^2+a)/(d*x^2+c))

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Rubi [A]  time = 0.31, antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1960, 463, 455, 385, 208} \[ -\frac {\left (-11 a^2 d^2+2 a b c d+b^2 c^2\right ) (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{16 a^2 c^3 \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {\sqrt {e} \left (5 a^2 d^2+2 a b c d+b^2 c^2\right ) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{16 a^{5/2} c^{7/2}}+\frac {e^2 (b c-a d)^3 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{6 a c^2 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^3}+\frac {(3 a d+b c) (b c-a d)^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 a c^3 \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[(e*(a + b*x^2))/(c + d*x^2)]/x^7,x]

[Out]

((b*c - a*d)^2*(b*c + 3*a*d)*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(8*a*c^3*(a - (c*(a + b*x^2))/(c + d*x^2))^2)
- ((b*c - a*d)*(b^2*c^2 + 2*a*b*c*d - 11*a^2*d^2)*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(16*a^2*c^3*(a - (c*(a +
b*x^2))/(c + d*x^2))) + ((b*c - a*d)^3*e^2*((e*(a + b*x^2))/(c + d*x^2))^(3/2))/(6*a*c^2*(a*e - (c*e*(a + b*x^
2))/(c + d*x^2))^3) - ((b*c - a*d)*(b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*Sqrt[e]*ArcTanh[(Sqrt[c]*Sqrt[(e*(a + b*x
^2))/(c + d*x^2)])/(Sqrt[a]*Sqrt[e])])/(16*a^(5/2)*c^(7/2))

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 455

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[(a + b*x^2)^(p + 1)*E
xpandToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)]
- (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[
m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 463

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

Rule 1960

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> With[{q = Den
ominator[p]}, Dist[(q*e*(b*c - a*d))/n, Subst[Int[(x^(q*(p + 1) - 1)*(-(a*e) + c*x^q)^(Simplify[(m + 1)/n] - 1
))/(b*e - d*x^q)^(Simplify[(m + 1)/n] + 1), x], x, ((e*(a + b*x^n))/(c + d*x^n))^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, m, n}, x] && FractionQ[p] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^7} \, dx &=((b c-a d) e) \operatorname {Subst}\left (\int \frac {x^2 \left (b e-d x^2\right )^2}{\left (-a e+c x^2\right )^4} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )\\ &=\frac {(b c-a d)^3 e^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{6 a c^2 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^3}+\frac {(b c-a d) \operatorname {Subst}\left (\int \frac {x^2 \left (-3 \left (2 b^2 c^2 e^2-(b c e-a d e)^2\right )+6 a c d^2 e x^2\right )}{\left (-a e+c x^2\right )^3} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{6 a c^2}\\ &=\frac {(b c-a d)^2 (b c+3 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 a c^3 \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )^2}+\frac {(b c-a d)^3 e^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{6 a c^2 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^3}-\frac {(b c-a d) \operatorname {Subst}\left (\int \frac {3 c (b c-a d) (b c+3 a d) e^2-24 a c^2 d^2 e x^2}{\left (-a e+c x^2\right )^2} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{24 a c^4}\\ &=\frac {(b c-a d)^2 (b c+3 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 a c^3 \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac {(b c-a d) \left (b^2 c^2+2 a b c d-11 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{16 a^2 c^3 \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )}+\frac {(b c-a d)^3 e^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{6 a c^2 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^3}+\frac {\left ((b c-a d) \left (b^2 c^2+2 a b c d+5 a^2 d^2\right ) e\right ) \operatorname {Subst}\left (\int \frac {1}{-a e+c x^2} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{16 a^2 c^3}\\ &=\frac {(b c-a d)^2 (b c+3 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 a c^3 \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac {(b c-a d) \left (b^2 c^2+2 a b c d-11 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{16 a^2 c^3 \left (a-\frac {c \left (a+b x^2\right )}{c+d x^2}\right )}+\frac {(b c-a d)^3 e^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{6 a c^2 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^3}-\frac {(b c-a d) \left (b^2 c^2+2 a b c d+5 a^2 d^2\right ) \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{16 a^{5/2} c^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 222, normalized size = 0.70 \[ \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\sqrt {a} \sqrt {c} \sqrt {a+b x^2} \sqrt {c+d x^2} \left (a^2 \left (-8 c^2+10 c d x^2-15 d^2 x^4\right )-2 a b c x^2 \left (c-2 d x^2\right )+3 b^2 c^2 x^4\right )-3 x^6 \left (-5 a^3 d^3+3 a^2 b c d^2+a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )\right )}{48 a^{5/2} c^{7/2} x^6 \sqrt {a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(e*(a + b*x^2))/(c + d*x^2)]/x^7,x]

[Out]

(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[c + d*x^2]*(Sqrt[a]*Sqrt[c]*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(3*b^2*c^2
*x^4 - 2*a*b*c*x^2*(c - 2*d*x^2) + a^2*(-8*c^2 + 10*c*d*x^2 - 15*d^2*x^4)) - 3*(b^3*c^3 + a*b^2*c^2*d + 3*a^2*
b*c*d^2 - 5*a^3*d^3)*x^6*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])]))/(48*a^(5/2)*c^(7/2)*x^
6*Sqrt[a + b*x^2])

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fricas [A]  time = 3.20, size = 561, normalized size = 1.76 \[ \left [-\frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} x^{6} \sqrt {\frac {e}{a c}} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} e x^{4} + 8 \, a^{2} c^{2} e + 8 \, {\left (a b c^{2} + a^{2} c d\right )} e x^{2} + 4 \, {\left (2 \, a^{2} c^{3} + {\left (a b c^{2} d + a^{2} c d^{2}\right )} x^{4} + {\left (a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {\frac {e}{a c}}}{x^{4}}\right ) - 4 \, {\left ({\left (3 \, b^{2} c^{2} d + 4 \, a b c d^{2} - 15 \, a^{2} d^{3}\right )} x^{6} - 8 \, a^{2} c^{3} + {\left (3 \, b^{2} c^{3} + 2 \, a b c^{2} d - 5 \, a^{2} c d^{2}\right )} x^{4} - 2 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{192 \, a^{2} c^{3} x^{6}}, \frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} x^{6} \sqrt {-\frac {e}{a c}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {-\frac {e}{a c}}}{2 \, {\left (b e x^{2} + a e\right )}}\right ) + 2 \, {\left ({\left (3 \, b^{2} c^{2} d + 4 \, a b c d^{2} - 15 \, a^{2} d^{3}\right )} x^{6} - 8 \, a^{2} c^{3} + {\left (3 \, b^{2} c^{3} + 2 \, a b c^{2} d - 5 \, a^{2} c d^{2}\right )} x^{4} - 2 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{96 \, a^{2} c^{3} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*(b*x^2+a)/(d*x^2+c))^(1/2)/x^7,x, algorithm="fricas")

[Out]

[-1/192*(3*(b^3*c^3 + a*b^2*c^2*d + 3*a^2*b*c*d^2 - 5*a^3*d^3)*x^6*sqrt(e/(a*c))*log(((b^2*c^2 + 6*a*b*c*d + a
^2*d^2)*e*x^4 + 8*a^2*c^2*e + 8*(a*b*c^2 + a^2*c*d)*e*x^2 + 4*(2*a^2*c^3 + (a*b*c^2*d + a^2*c*d^2)*x^4 + (a*b*
c^3 + 3*a^2*c^2*d)*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c))*sqrt(e/(a*c)))/x^4) - 4*((3*b^2*c^2*d + 4*a*b*c*d^2
- 15*a^2*d^3)*x^6 - 8*a^2*c^3 + (3*b^2*c^3 + 2*a*b*c^2*d - 5*a^2*c*d^2)*x^4 - 2*(a*b*c^3 - a^2*c^2*d)*x^2)*sqr
t((b*e*x^2 + a*e)/(d*x^2 + c)))/(a^2*c^3*x^6), 1/96*(3*(b^3*c^3 + a*b^2*c^2*d + 3*a^2*b*c*d^2 - 5*a^3*d^3)*x^6
*sqrt(-e/(a*c))*arctan(1/2*((b*c + a*d)*x^2 + 2*a*c)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c))*sqrt(-e/(a*c))/(b*e*x^2
+ a*e)) + 2*((3*b^2*c^2*d + 4*a*b*c*d^2 - 15*a^2*d^3)*x^6 - 8*a^2*c^3 + (3*b^2*c^3 + 2*a*b*c^2*d - 5*a^2*c*d^
2)*x^4 - 2*(a*b*c^3 - a^2*c^2*d)*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(a^2*c^3*x^6)]

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giac [B]  time = 0.98, size = 1076, normalized size = 3.38 \[ \frac {1}{48} \, {\left (\frac {3 \, {\left (b^{3} c^{3} e + a b^{2} c^{2} d e + 3 \, a^{2} b c d^{2} e - 5 \, a^{3} d^{3} e\right )} \arctan \left (-\frac {\sqrt {b d} x^{2} e^{\frac {1}{2}} - \sqrt {b d x^{4} e + b c x^{2} e + a d x^{2} e + a c e}}{\sqrt {-a c e}}\right )}{\sqrt {-a c e} a^{2} c^{3}} - \frac {3 \, {\left (\sqrt {b d} x^{2} e^{\frac {1}{2}} - \sqrt {b d x^{4} e + b c x^{2} e + a d x^{2} e + a c e}\right )} a^{2} b^{3} c^{5} e^{3} + 51 \, {\left (\sqrt {b d} x^{2} e^{\frac {1}{2}} - \sqrt {b d x^{4} e + b c x^{2} e + a d x^{2} e + a c e}\right )} a^{3} b^{2} c^{4} d e^{3} + 105 \, {\left (\sqrt {b d} x^{2} e^{\frac {1}{2}} - \sqrt {b d x^{4} e + b c x^{2} e + a d x^{2} e + a c e}\right )} a^{4} b c^{3} d^{2} e^{3} + 33 \, {\left (\sqrt {b d} x^{2} e^{\frac {1}{2}} - \sqrt {b d x^{4} e + b c x^{2} e + a d x^{2} e + a c e}\right )} a^{5} c^{2} d^{3} e^{3} + 8 \, {\left (\sqrt {b d} x^{2} e^{\frac {1}{2}} - \sqrt {b d x^{4} e + b c x^{2} e + a d x^{2} e + a c e}\right )}^{3} a b^{3} c^{4} e^{2} + 72 \, {\left (\sqrt {b d} x^{2} e^{\frac {1}{2}} - \sqrt {b d x^{4} e + b c x^{2} e + a d x^{2} e + a c e}\right )}^{3} a^{2} b^{2} c^{3} d e^{2} + 24 \, {\left (\sqrt {b d} x^{2} e^{\frac {1}{2}} - \sqrt {b d x^{4} e + b c x^{2} e + a d x^{2} e + a c e}\right )}^{3} a^{3} b c^{2} d^{2} e^{2} - 40 \, {\left (\sqrt {b d} x^{2} e^{\frac {1}{2}} - \sqrt {b d x^{4} e + b c x^{2} e + a d x^{2} e + a c e}\right )}^{3} a^{4} c d^{3} e^{2} - 3 \, {\left (\sqrt {b d} x^{2} e^{\frac {1}{2}} - \sqrt {b d x^{4} e + b c x^{2} e + a d x^{2} e + a c e}\right )}^{5} b^{3} c^{3} e - 3 \, {\left (\sqrt {b d} x^{2} e^{\frac {1}{2}} - \sqrt {b d x^{4} e + b c x^{2} e + a d x^{2} e + a c e}\right )}^{5} a b^{2} c^{2} d e - 9 \, {\left (\sqrt {b d} x^{2} e^{\frac {1}{2}} - \sqrt {b d x^{4} e + b c x^{2} e + a d x^{2} e + a c e}\right )}^{5} a^{2} b c d^{2} e + 15 \, {\left (\sqrt {b d} x^{2} e^{\frac {1}{2}} - \sqrt {b d x^{4} e + b c x^{2} e + a d x^{2} e + a c e}\right )}^{5} a^{3} d^{3} e + 16 \, \sqrt {b d} a^{4} b c^{4} d e^{\frac {7}{2}} + 48 \, \sqrt {b d} a^{5} c^{3} d^{2} e^{\frac {7}{2}} + 48 \, {\left (\sqrt {b d} x^{2} e^{\frac {1}{2}} - \sqrt {b d x^{4} e + b c x^{2} e + a d x^{2} e + a c e}\right )}^{2} \sqrt {b d} a^{2} b^{2} c^{4} e^{\frac {5}{2}} + 144 \, {\left (\sqrt {b d} x^{2} e^{\frac {1}{2}} - \sqrt {b d x^{4} e + b c x^{2} e + a d x^{2} e + a c e}\right )}^{2} \sqrt {b d} a^{3} b c^{3} d e^{\frac {5}{2}}}{{\left (a c e - {\left (\sqrt {b d} x^{2} e^{\frac {1}{2}} - \sqrt {b d x^{4} e + b c x^{2} e + a d x^{2} e + a c e}\right )}^{2}\right )}^{3} a^{2} c^{3}}\right )} \mathrm {sgn}\left (d x^{2} + c\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*(b*x^2+a)/(d*x^2+c))^(1/2)/x^7,x, algorithm="giac")

[Out]

1/48*(3*(b^3*c^3*e + a*b^2*c^2*d*e + 3*a^2*b*c*d^2*e - 5*a^3*d^3*e)*arctan(-(sqrt(b*d)*x^2*e^(1/2) - sqrt(b*d*
x^4*e + b*c*x^2*e + a*d*x^2*e + a*c*e))/sqrt(-a*c*e))/(sqrt(-a*c*e)*a^2*c^3) - (3*(sqrt(b*d)*x^2*e^(1/2) - sqr
t(b*d*x^4*e + b*c*x^2*e + a*d*x^2*e + a*c*e))*a^2*b^3*c^5*e^3 + 51*(sqrt(b*d)*x^2*e^(1/2) - sqrt(b*d*x^4*e + b
*c*x^2*e + a*d*x^2*e + a*c*e))*a^3*b^2*c^4*d*e^3 + 105*(sqrt(b*d)*x^2*e^(1/2) - sqrt(b*d*x^4*e + b*c*x^2*e + a
*d*x^2*e + a*c*e))*a^4*b*c^3*d^2*e^3 + 33*(sqrt(b*d)*x^2*e^(1/2) - sqrt(b*d*x^4*e + b*c*x^2*e + a*d*x^2*e + a*
c*e))*a^5*c^2*d^3*e^3 + 8*(sqrt(b*d)*x^2*e^(1/2) - sqrt(b*d*x^4*e + b*c*x^2*e + a*d*x^2*e + a*c*e))^3*a*b^3*c^
4*e^2 + 72*(sqrt(b*d)*x^2*e^(1/2) - sqrt(b*d*x^4*e + b*c*x^2*e + a*d*x^2*e + a*c*e))^3*a^2*b^2*c^3*d*e^2 + 24*
(sqrt(b*d)*x^2*e^(1/2) - sqrt(b*d*x^4*e + b*c*x^2*e + a*d*x^2*e + a*c*e))^3*a^3*b*c^2*d^2*e^2 - 40*(sqrt(b*d)*
x^2*e^(1/2) - sqrt(b*d*x^4*e + b*c*x^2*e + a*d*x^2*e + a*c*e))^3*a^4*c*d^3*e^2 - 3*(sqrt(b*d)*x^2*e^(1/2) - sq
rt(b*d*x^4*e + b*c*x^2*e + a*d*x^2*e + a*c*e))^5*b^3*c^3*e - 3*(sqrt(b*d)*x^2*e^(1/2) - sqrt(b*d*x^4*e + b*c*x
^2*e + a*d*x^2*e + a*c*e))^5*a*b^2*c^2*d*e - 9*(sqrt(b*d)*x^2*e^(1/2) - sqrt(b*d*x^4*e + b*c*x^2*e + a*d*x^2*e
+ a*c*e))^5*a^2*b*c*d^2*e + 15*(sqrt(b*d)*x^2*e^(1/2) - sqrt(b*d*x^4*e + b*c*x^2*e + a*d*x^2*e + a*c*e))^5*a^
3*d^3*e + 16*sqrt(b*d)*a^4*b*c^4*d*e^(7/2) + 48*sqrt(b*d)*a^5*c^3*d^2*e^(7/2) + 48*(sqrt(b*d)*x^2*e^(1/2) - sq
rt(b*d*x^4*e + b*c*x^2*e + a*d*x^2*e + a*c*e))^2*sqrt(b*d)*a^2*b^2*c^4*e^(5/2) + 144*(sqrt(b*d)*x^2*e^(1/2) -
sqrt(b*d*x^4*e + b*c*x^2*e + a*d*x^2*e + a*c*e))^2*sqrt(b*d)*a^3*b*c^3*d*e^(5/2))/((a*c*e - (sqrt(b*d)*x^2*e^(
1/2) - sqrt(b*d*x^4*e + b*c*x^2*e + a*d*x^2*e + a*c*e))^2)^3*a^2*c^3))*sgn(d*x^2 + c)

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maple [B]  time = 0.08, size = 849, normalized size = 2.67 \[ -\frac {\sqrt {\frac {\left (b \,x^{2}+a \right ) e}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (-15 a^{4} c \,d^{3} x^{6} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}{x^{2}}\right )+9 a^{3} b \,c^{2} d^{2} x^{6} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}{x^{2}}\right )+3 a^{2} b^{2} c^{3} d \,x^{6} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}{x^{2}}\right )+3 a \,b^{3} c^{4} x^{6} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}{x^{2}}\right )-66 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a^{2} b \,d^{3} x^{8}-24 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a \,b^{2} c \,d^{2} x^{8}-6 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, b^{3} c^{2} d \,x^{8}-66 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a^{3} d^{3} x^{6}-54 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a^{2} b c \,d^{2} x^{6}-18 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a \,b^{2} c^{2} d \,x^{6}-6 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, b^{3} c^{3} x^{6}+66 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, a^{2} d^{2} x^{4}+24 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, a b c d \,x^{4}+6 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, b^{2} c^{2} x^{4}-36 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, a^{2} c d \,x^{2}-12 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, a b \,c^{2} x^{2}+16 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, a^{2} c^{2}\right )}{96 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {a c}\, a^{3} c^{4} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x^2+a)/(d*x^2+c)*e)^(1/2)/x^7,x)

[Out]

-1/96*((b*x^2+a)/(d*x^2+c)*e)^(1/2)*(d*x^2+c)*(-66*b*d^3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*x^8*a^2*(a*c)^(1/
2)-24*b^2*d^2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*x^8*a*c*(a*c)^(1/2)-6*b^3*d*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1
/2)*x^8*c^2*(a*c)^(1/2)-15*a^4*ln((a*d*x^2+b*c*x^2+2*a*c+2*(a*c)^(1/2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2))/x^
2)*d^3*c*x^6+9*ln((a*d*x^2+b*c*x^2+2*a*c+2*(a*c)^(1/2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2))/x^2)*d^2*b*a^3*c^2
*x^6+3*ln((a*d*x^2+b*c*x^2+2*a*c+2*(a*c)^(1/2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2))/x^2)*d*b^2*a^2*c^3*x^6+3*c
^4*ln((a*d*x^2+b*c*x^2+2*a*c+2*(a*c)^(1/2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2))/x^2)*b^3*a*x^6-66*(b*d*x^4+a*d
*x^2+b*c*x^2+a*c)^(1/2)*d^3*a^3*x^6*(a*c)^(1/2)-54*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*d^2*b*a^2*c*x^6*(a*c)^(
1/2)-18*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*b^2*d*a*c^2*x^6*(a*c)^(1/2)-6*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*
b^3*c^3*x^6*(a*c)^(1/2)+66*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(3/2)*d^2*a^2*x^4*(a*c)^(1/2)+24*(b*d*x^4+a*d*x^2+b*c
*x^2+a*c)^(3/2)*d*b*a*c*x^4*(a*c)^(1/2)+6*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(3/2)*b^2*c^2*x^4*(a*c)^(1/2)-36*(b*d*
x^4+a*d*x^2+b*c*x^2+a*c)^(3/2)*d*a^2*c*x^2*(a*c)^(1/2)-12*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(3/2)*b*a*c^2*x^2*(a*c
)^(1/2)+16*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(3/2)*a^2*c^2*(a*c)^(1/2))/((d*x^2+c)*(b*x^2+a))^(1/2)/c^4/a^3/x^6/(a
*c)^(1/2)

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maxima [A]  time = 2.29, size = 410, normalized size = 1.29 \[ -\frac {1}{96} \, e {\left (\frac {2 \, {\left (3 \, {\left (b^{3} c^{5} + a b^{2} c^{4} d - 13 \, a^{2} b c^{3} d^{2} + 11 \, a^{3} c^{2} d^{3}\right )} \left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {5}{2}} - 8 \, {\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d - 3 \, a^{3} b c^{2} d^{2} + 5 \, a^{4} c d^{3}\right )} \left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}} e - 3 \, {\left (a^{2} b^{3} c^{3} + a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - 5 \, a^{5} d^{3}\right )} \sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}} e^{2}\right )}}{a^{5} c^{3} e^{3} - \frac {3 \, {\left (b x^{2} + a\right )} a^{4} c^{4} e^{3}}{d x^{2} + c} + \frac {3 \, {\left (b x^{2} + a\right )}^{2} a^{3} c^{5} e^{3}}{{\left (d x^{2} + c\right )}^{2}} - \frac {{\left (b x^{2} + a\right )}^{3} a^{2} c^{6} e^{3}}{{\left (d x^{2} + c\right )}^{3}}} - \frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \log \left (\frac {c \sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}} - \sqrt {a c e}}{c \sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}} + \sqrt {a c e}}\right )}{\sqrt {a c e} a^{2} c^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*(b*x^2+a)/(d*x^2+c))^(1/2)/x^7,x, algorithm="maxima")

[Out]

-1/96*e*(2*(3*(b^3*c^5 + a*b^2*c^4*d - 13*a^2*b*c^3*d^2 + 11*a^3*c^2*d^3)*((b*x^2 + a)*e/(d*x^2 + c))^(5/2) -
8*(a*b^3*c^4 - 3*a^2*b^2*c^3*d - 3*a^3*b*c^2*d^2 + 5*a^4*c*d^3)*((b*x^2 + a)*e/(d*x^2 + c))^(3/2)*e - 3*(a^2*b
^3*c^3 + a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - 5*a^5*d^3)*sqrt((b*x^2 + a)*e/(d*x^2 + c))*e^2)/(a^5*c^3*e^3 - 3*(b*x
^2 + a)*a^4*c^4*e^3/(d*x^2 + c) + 3*(b*x^2 + a)^2*a^3*c^5*e^3/(d*x^2 + c)^2 - (b*x^2 + a)^3*a^2*c^6*e^3/(d*x^2
+ c)^3) - 3*(b^3*c^3 + a*b^2*c^2*d + 3*a^2*b*c*d^2 - 5*a^3*d^3)*log((c*sqrt((b*x^2 + a)*e/(d*x^2 + c)) - sqrt
(a*c*e))/(c*sqrt((b*x^2 + a)*e/(d*x^2 + c)) + sqrt(a*c*e)))/(sqrt(a*c*e)*a^2*c^3))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}}}{x^7} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((e*(a + b*x^2))/(c + d*x^2))^(1/2)/x^7,x)

[Out]

int(((e*(a + b*x^2))/(c + d*x^2))^(1/2)/x^7, 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))**(1/2)/x**7,x)

[Out]

Timed out

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