3.3.69 \(\int \frac {\sqrt {\frac {e (a+b x^2)}{c+d x^2}}}{x^7} \, dx\) [269]

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

[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 = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1981, 1980, 474, 466, 393, 214} \begin {gather*} -\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} \end {gather*}

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 214

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

Rule 393

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 466

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 474

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 1980

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)))/((c_) + (d_.)*(x_)))^(p_), x_Symbol] :> With[{q = Denominator[p]}
, Dist[q*e*(b*c - a*d), Subst[Int[x^(q*(p + 1) - 1)*(((-a)*e + c*x^q)^m/(b*e - d*x^q)^(m + 2)), x], x, (e*((a
+ b*x)/(c + d*x)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e, m}, x] && FractionQ[p] && IntegerQ[m]

Rule 1981

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Dist[1/n, Sub
st[Int[x^(Simplify[(m + 1)/n] - 1)*(e*((a + b*x)/(c + d*x)))^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n,
p}, x] && 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) \text {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) \text {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) \text {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 ) \text {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 = 1.50, size = 222, normalized size = 0.70 \begin {gather*} \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2} \left (\sqrt {a} \sqrt {c} \sqrt {a+b x^2} \sqrt {c+d x^2} \left (3 b^2 c^2 x^4-2 a b c x^2 \left (c-2 d x^2\right )+a^2 \left (-8 c^2+10 c d x^2-15 d^2 x^4\right )\right )-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 \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}} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(848\) vs. \(2(290)=580\).
time = 0.11, size = 849, normalized size = 2.67

method	result	size
risch	\(-\frac {\left (d \,x^{2}+c \right ) \left (15 d^{2} a^{2} x^{4}-4 a b c d \,x^{4}-3 b^{2} c^{2} x^{4}-10 a^{2} d c \,x^{2}+2 a b \,c^{2} x^{2}+8 a^{2} c^{2}\right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{48 c^{3} x^{6} a^{2}}+\frac {\left (\frac {5 a \ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) d^{3}}{32 c^{3} \sqrt {a c e}}-\frac {3 \ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) b \,d^{2}}{32 c^{2} \sqrt {a c e}}-\frac {\ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) d \,b^{2}}{32 a c \sqrt {a c e}}-\frac {\ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) b^{3}}{32 a^{2} \sqrt {a c e}}\right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{b \,x^{2}+a}\)	\(447\)
default	\(\frac {\sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (66 b \,d^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, x^{8} a^{2} \sqrt {a c}+24 b^{2} d^{2} \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, x^{8} a c \sqrt {a c}+6 b^{3} d \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, x^{8} c^{2} \sqrt {a c}+15 a^{4} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) d^{3} c \,x^{6}-9 \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) d^{2} b \,a^{3} c^{2} x^{6}-3 \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) d \,b^{2} a^{2} c^{3} x^{6}-3 c^{4} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) b^{3} a \,x^{6}+66 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, d^{3} a^{3} x^{6} \sqrt {a c}+54 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, d^{2} b \,a^{2} c \,x^{6} \sqrt {a c}+18 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, b^{2} d a \,c^{2} x^{6} \sqrt {a c}+6 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, b^{3} c^{3} x^{6} \sqrt {a c}-66 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} d^{2} a^{2} x^{4} \sqrt {a c}-24 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} d b a c \,x^{4} \sqrt {a c}-6 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} b^{2} c^{2} x^{4} \sqrt {a c}+36 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} d \,a^{2} c \,x^{2} \sqrt {a c}+12 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} b a \,c^{2} x^{2} \sqrt {a c}-16 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} a^{2} c^{2} \sqrt {a c}\right )}{96 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, c^{4} a^{3} x^{6} \sqrt {a c}}\)	\(849\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(e*(b*x^2+a)/(d*x^2+c))^(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)^(1/2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+2*a*c)/x^2)
*d^3*c*x^6-9*ln((a*d*x^2+b*c*x^2+2*(a*c)^(1/2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+2*a*c)/x^2)*d^2*b*a^3*c^2*x
^6-3*ln((a*d*x^2+b*c*x^2+2*(a*c)^(1/2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+2*a*c)/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)^(1/2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+2*a*c)/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 = 0.54, size = 387, normalized size = 1.22 \begin {gather*} -\frac {1}{96} \, {\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 {b x^{2} + a}{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 {b x^{2} + a}{d x^{2} + c}\right )^{\frac {3}{2}} - 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 {b x^{2} + a}{d x^{2} + c}}\right )}}{a^{5} c^{3} - \frac {3 \, {\left (b x^{2} + a\right )} a^{4} c^{4}}{d x^{2} + c} + \frac {3 \, {\left (b x^{2} + a\right )}^{2} a^{3} c^{5}}{{\left (d x^{2} + c\right )}^{2}} - \frac {{\left (b x^{2} + a\right )}^{3} a^{2} c^{6}}{{\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 {b x^{2} + a}{d x^{2} + c}} - \sqrt {a c}}{c \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} + \sqrt {a c}}\right )}{\sqrt {a c} a^{2} c^{3}}\right )} e^{\frac {1}{2}} \end {gather*}

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

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Fricas [A]
time = 1.24, size = 554, normalized size = 1.74 \begin {gather*} \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 )} \sqrt {a c} x^{6} e^{\frac {1}{2}} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b c d + a d^{2}\right )} x^{4} + 2 \, a c^{2} + {\left (b c^{2} + 3 \, a c d\right )} x^{2}\right )} \sqrt {a c} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}}{x^{4}}\right ) + 4 \, {\left (8 \, a^{3} c^{4} - {\left (3 \, a b^{2} c^{3} d + 4 \, a^{2} b c^{2} d^{2} - 15 \, a^{3} c d^{3}\right )} x^{6} - {\left (3 \, a b^{2} c^{4} + 2 \, a^{2} b c^{3} d - 5 \, a^{3} c^{2} d^{2}\right )} x^{4} + 2 \, {\left (a^{2} b c^{4} - a^{3} c^{3} d\right )} x^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} e^{\frac {1}{2}}}{192 \, a^{3} c^{4} 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 )} \sqrt {-a c} x^{6} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {-a c} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}}{2 \, {\left (a b c x^{2} + a^{2} c\right )}}\right ) e^{\frac {1}{2}} - 2 \, {\left (8 \, a^{3} c^{4} - {\left (3 \, a b^{2} c^{3} d + 4 \, a^{2} b c^{2} d^{2} - 15 \, a^{3} c d^{3}\right )} x^{6} - {\left (3 \, a b^{2} c^{4} + 2 \, a^{2} b c^{3} d - 5 \, a^{3} c^{2} d^{2}\right )} x^{4} + 2 \, {\left (a^{2} b c^{4} - a^{3} c^{3} d\right )} x^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} e^{\frac {1}{2}}}{96 \, a^{3} c^{4} x^{6}}\right ] \end {gather*}

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 871 vs. \(2 (287) = 574\).
time = 6.96, size = 871, normalized size = 2.74 \begin {gather*} \frac {1}{48} \, {\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 )} \arctan \left (-\frac {\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}}{\sqrt {-a c}}\right )}{\sqrt {-a c} a^{2} c^{3}} - \frac {3 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{5} b^{3} c^{3} - 8 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{3} a b^{3} c^{4} - 3 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} a^{2} b^{3} c^{5} + 3 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{5} a b^{2} c^{2} d - 72 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{3} a^{2} b^{2} c^{3} d - 51 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} a^{3} b^{2} c^{4} d + 9 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{5} a^{2} b c d^{2} - 24 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{3} a^{3} b c^{2} d^{2} - 105 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} a^{4} b c^{3} d^{2} - 15 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{5} a^{3} d^{3} + 40 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{3} a^{4} c d^{3} - 33 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} a^{5} c^{2} d^{3} - 48 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{2} \sqrt {b d} a^{2} b^{2} c^{4} - 144 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{2} \sqrt {b d} a^{3} b c^{3} d - 16 \, \sqrt {b d} a^{4} b c^{4} d - 48 \, \sqrt {b d} a^{5} c^{3} d^{2}}{{\left ({\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{2} - a c\right )}^{3} a^{2} c^{3}}\right )} e^{\frac {1}{2}} \mathrm {sgn}\left (d x^{2} + c\right ) \end {gather*}

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

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

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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