3.3.82 \(\int \frac {(\frac {e (a+b x^2)}{c+d x^2})^{3/2}}{x^7} \, dx\) [282]

Optimal. Leaf size=366 \[ \frac {d^2 (b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c^4}+\frac {(b c-a d)^3 e^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2}}{6 a c^2 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^3}+\frac {(b c-a d)^2 (b c+11 a d) e^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{24 c^4 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac {(b c-a d) \left (5 b^2 c^2+50 a b c d-79 a^2 d^2\right ) e^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{48 a c^4 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}+\frac {(b c-a d) \left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) e^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{16 a^{3/2} c^{9/2}} \]

[Out]

1/6*(-a*d+b*c)^3*e^2*(e*(b*x^2+a)/(d*x^2+c))^(5/2)/a/c^2/(a*e-c*e*(b*x^2+a)/(d*x^2+c))^3+1/16*(-a*d+b*c)*(-35*
a^2*d^2+10*a*b*c*d+b^2*c^2)*e^(3/2)*arctanh(c^(1/2)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/a^(1/2)/e^(1/2))/a^(3/2)/c^(
9/2)+d^2*(-a*d+b*c)*e*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/c^4+1/24*(-a*d+b*c)^2*(11*a*d+b*c)*e^3*(e*(b*x^2+a)/(d*x^2
+c))^(1/2)/c^4/(a*e-c*e*(b*x^2+a)/(d*x^2+c))^2-1/48*(-a*d+b*c)*(-79*a^2*d^2+50*a*b*c*d+5*b^2*c^2)*e^2*(e*(b*x^
2+a)/(d*x^2+c))^(1/2)/a/c^4/(a*e-c*e*(b*x^2+a)/(d*x^2+c))

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Rubi [A]
time = 0.37, antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1981, 1980, 474, 466, 1171, 396, 214} \begin {gather*} -\frac {e^2 \left (-79 a^2 d^2+50 a b c d+5 b^2 c^2\right ) (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{48 a c^4 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}+\frac {e^{3/2} \left (-35 a^2 d^2+10 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^{3/2} c^{9/2}}+\frac {d^2 e (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c^4}+\frac {e^3 (11 a d+b c) (b c-a d)^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{24 c^4 \left (a e-\frac {c e \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 )^{5/2}}{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[((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^7,x]

[Out]

(d^2*(b*c - a*d)*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/c^4 + ((b*c - a*d)^3*e^2*((e*(a + b*x^2))/(c + d*x^2))^(
5/2))/(6*a*c^2*(a*e - (c*e*(a + b*x^2))/(c + d*x^2))^3) + ((b*c - a*d)^2*(b*c + 11*a*d)*e^3*Sqrt[(e*(a + b*x^2
))/(c + d*x^2)])/(24*c^4*(a*e - (c*e*(a + b*x^2))/(c + d*x^2))^2) - ((b*c - a*d)*(5*b^2*c^2 + 50*a*b*c*d - 79*
a^2*d^2)*e^2*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(48*a*c^4*(a*e - (c*e*(a + b*x^2))/(c + d*x^2))) + ((b*c - a*d
)*(b^2*c^2 + 10*a*b*c*d - 35*a^2*d^2)*e^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(Sqrt[a]*Sqr
t[e])])/(16*a^(3/2)*c^(9/2))

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 396

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(
p + 1) + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 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 1171

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1
)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -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 {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^7} \, dx &=((b c-a d) e) \text {Subst}\left (\int \frac {x^4 \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 )^{5/2}}{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^4 \left (-6 b^2 c^2 e^2+5 (b c e-a d e)^2+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)^3 e^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2}}{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)^2 (b c+11 a d) e^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{24 c^4 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac {(b c-a d) \text {Subst}\left (\int \frac {a c (b c-a d) (b c+11 a d) e^3+4 c^2 (b c-a d) (b c+11 a d) e^2 x^2-24 a c^3 d^2 e x^4}{\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^5}\\ &=\frac {(b c-a d)^3 e^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2}}{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)^2 (b c+11 a d) e^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{24 c^4 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac {(b c-a d) \left (5 b^2 c^2+50 a b c d-79 a^2 d^2\right ) e^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{48 a c^4 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {(b c-a d) \text {Subst}\left (\int \frac {3 a c \left (b^2 c^2+10 a b c d-19 a^2 d^2\right ) e^3-48 a^2 c^2 d^2 e^2 x^2}{-a e+c x^2} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{48 a^2 c^5 e}\\ &=\frac {d^2 (b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c^4}+\frac {(b c-a d)^3 e^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2}}{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)^2 (b c+11 a d) e^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{24 c^4 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac {(b c-a d) \left (5 b^2 c^2+50 a b c d-79 a^2 d^2\right ) e^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{48 a c^4 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {\left ((b c-a d) \left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) e^2\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 c^4}\\ &=\frac {d^2 (b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c^4}+\frac {(b c-a d)^3 e^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{5/2}}{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)^2 (b c+11 a d) e^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{24 c^4 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac {(b c-a d) \left (5 b^2 c^2+50 a b c d-79 a^2 d^2\right ) e^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{48 a c^4 \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}+\frac {(b c-a d) \left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) e^{3/2} \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^{3/2} c^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 4.94, size = 245, normalized size = 0.67 \begin {gather*} \frac {e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\sqrt {a} \sqrt {c} \sqrt {a+b x^2} \left (3 b^2 c^2 x^4 \left (c+d x^2\right )+2 a b c x^2 \left (7 c^2-19 c d x^2-50 d^2 x^4\right )+a^2 \left (8 c^3-14 c^2 d x^2+35 c d^2 x^4+105 d^3 x^6\right )\right )+3 \left (b^3 c^3+9 a b^2 c^2 d-45 a^2 b c d^2+35 a^3 d^3\right ) x^6 \sqrt {c+d x^2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )\right )}{48 a^{3/2} c^{9/2} x^6 \sqrt {a+b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^7,x]

[Out]

(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(-(Sqrt[a]*Sqrt[c]*Sqrt[a + b*x^2]*(3*b^2*c^2*x^4*(c + d*x^2) + 2*a*b*c*x
^2*(7*c^2 - 19*c*d*x^2 - 50*d^2*x^4) + a^2*(8*c^3 - 14*c^2*d*x^2 + 35*c*d^2*x^4 + 105*d^3*x^6))) + 3*(b^3*c^3
+ 9*a*b^2*c^2*d - 45*a^2*b*c*d^2 + 35*a^3*d^3)*x^6*Sqrt[c + d*x^2]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*
Sqrt[c + d*x^2])]))/(48*a^(3/2)*c^(9/2)*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. \(1497\) vs. \(2(336)=672\).
time = 0.15, size = 1498, normalized size = 4.09

method	result	size
risch	\(-\frac {\left (d \,x^{2}+c \right ) \left (57 d^{2} a^{2} x^{4}-52 a b c d \,x^{4}+3 b^{2} c^{2} x^{4}-22 a^{2} d c \,x^{2}+14 a b \,c^{2} x^{2}+8 a^{2} c^{2}\right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{48 c^{4} x^{6} a}+\frac {\left (\frac {35 a^{2} \ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) d^{3}}{32 c^{4} \sqrt {a c e}}-\frac {45 a \ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) b \,d^{2}}{32 c^{3} \sqrt {a c e}}+\frac {9 \ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) d \,b^{2}}{32 c^{2} \sqrt {a c e}}+\frac {\ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) b^{3}}{32 c a \sqrt {a c e}}-\frac {a^{2} d^{4} x^{2} b}{c^{4} \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {2 a \,d^{3} x^{2} b^{2}}{c^{3} \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {d^{2} x^{2} b^{3}}{c^{2} \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {a^{3} d^{4}}{c^{4} \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {2 a^{2} d^{3} b}{c^{3} \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {a \,d^{2} b^{2}}{c^{2} \left (a d -b c \right ) \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}\right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{b \,x^{2}+a}\)	\(759\)
default	\(\text {Expression too large to display}\)	\(1498\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*(b*x^2+a)/(d*x^2+c))^(3/2)/x^7,x,method=_RETURNVERBOSE)

[Out]

-1/96*(72*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(a*c)^(1/2)*a*b^2*c*d^3*x^10-216*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(
1/2)*(a*c)^(1/2)*a^2*b*c*d^3*x^8+138*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(a*c)^(1/2)*a*b^2*c^2*d^2*x^8-72*(b*d
*x^4+a*d*x^2+b*c*x^2+a*c)^(3/2)*(a*c)^(1/2)*a*b*c*d^2*x^6-42*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(a*c)^(1/2)*a
^2*b*c^2*d^2*x^6+66*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(a*c)^(1/2)*a*b^2*c^3*d*x^6-96*(a*c)^(1/2)*((d*x^2+c)*
(b*x^2+a))^(1/2)*a^2*b*c^2*d^2*x^6-60*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(3/2)*(a*c)^(1/2)*a*b*c^2*d*x^4+16*(b*d*x^
4+a*d*x^2+b*c*x^2+a*c)^(3/2)*(a*c)^(1/2)*a^2*c^3-174*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(a*c)^(1/2)*a^2*b*d^4
*x^10+6*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(a*c)^(1/2)*b^3*c^2*d^2*x^10+135*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)*a^3*b*c^2*d^3*x^8-27*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)*a^2*b^2*c^3*d^2*x^8-3*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)*a*b^3*c^4*d*x^8+12*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(a*c)^(1/2)*b^3*c^3
*d*x^8+135*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)*a^3*b*c^3*d^2*x^6
-27*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)*a^2*b^2*c^4*d*x^6-6*(b*d
*x^4+a*d*x^2+b*c*x^2+a*c)^(3/2)*(a*c)^(1/2)*b^2*c^2*d*x^6-174*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(a*c)^(1/2)*
a^3*c*d^3*x^6+96*(a*c)^(1/2)*((d*x^2+c)*(b*x^2+a))^(1/2)*a^3*c*d^3*x^6+114*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(3/2)
*(a*c)^(1/2)*a^2*c*d^2*x^4-44*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(3/2)*(a*c)^(1/2)*a^2*c^2*d*x^2+12*(b*d*x^4+a*d*x^
2+b*c*x^2+a*c)^(3/2)*(a*c)^(1/2)*a*b*c^3*x^2-105*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)*a^4*c*d^4*x^8-174*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(a*c)^(1/2)*a^3*d^4*x^8-105*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)*a^4*c^2*d^3*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)*a*b^3*c^5*x^6+174*(b*d*x^4+a*d*x^2+b*c*x^2+a*
c)^(3/2)*(a*c)^(1/2)*a^2*d^3*x^6+6*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(a*c)^(1/2)*b^3*c^4*x^6-6*(b*d*x^4+a*d*
x^2+b*c*x^2+a*c)^(3/2)*(a*c)^(1/2)*b^2*c^3*x^4)/a^2*(d*x^2+c)*(e*(b*x^2+a)/(d*x^2+c))^(3/2)/(a*c)^(1/2)/x^6/c^
5/(b*x^2+a)/((d*x^2+c)*(b*x^2+a))^(1/2)

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Maxima [A]
time = 0.52, size = 425, normalized size = 1.16 \begin {gather*} \frac {1}{96} \, {\left (\frac {2 \, {\left (3 \, {\left (b^{3} c^{5} - 23 \, a b^{2} c^{4} d + 51 \, a^{2} b c^{3} d^{2} - 29 \, 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} + 9 \, a^{2} b^{2} c^{3} d - 27 \, a^{3} b c^{2} d^{2} + 17 \, 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} + 9 \, a^{3} b^{2} c^{2} d - 29 \, a^{4} b c d^{2} + 19 \, a^{5} d^{3}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}\right )}}{a^{4} c^{4} - \frac {3 \, {\left (b x^{2} + a\right )} a^{3} c^{5}}{d x^{2} + c} + \frac {3 \, {\left (b x^{2} + a\right )}^{2} a^{2} c^{6}}{{\left (d x^{2} + c\right )}^{2}} - \frac {{\left (b x^{2} + a\right )}^{3} a c^{7}}{{\left (d x^{2} + c\right )}^{3}}} + \frac {96 \, {\left (b c d^{2} - a d^{3}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}}{c^{4}} - \frac {3 \, {\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d - 45 \, a^{2} b c d^{2} + 35 \, 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 c^{4}}\right )} e^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*(b*x^2+a)/(d*x^2+c))^(3/2)/x^7,x, algorithm="maxima")

[Out]

1/96*(2*(3*(b^3*c^5 - 23*a*b^2*c^4*d + 51*a^2*b*c^3*d^2 - 29*a^3*c^2*d^3)*((b*x^2 + a)/(d*x^2 + c))^(5/2) + 8*
(a*b^3*c^4 + 9*a^2*b^2*c^3*d - 27*a^3*b*c^2*d^2 + 17*a^4*c*d^3)*((b*x^2 + a)/(d*x^2 + c))^(3/2) - 3*(a^2*b^3*c
^3 + 9*a^3*b^2*c^2*d - 29*a^4*b*c*d^2 + 19*a^5*d^3)*sqrt((b*x^2 + a)/(d*x^2 + c)))/(a^4*c^4 - 3*(b*x^2 + a)*a^
3*c^5/(d*x^2 + c) + 3*(b*x^2 + a)^2*a^2*c^6/(d*x^2 + c)^2 - (b*x^2 + a)^3*a*c^7/(d*x^2 + c)^3) + 96*(b*c*d^2 -
a*d^3)*sqrt((b*x^2 + a)/(d*x^2 + c))/c^4 - 3*(b^3*c^3 + 9*a*b^2*c^2*d - 45*a^2*b*c*d^2 + 35*a^3*d^3)*log((c*s
qrt((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*c^4))*e^
(3/2)

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Fricas [A]
time = 4.46, size = 552, normalized size = 1.51 \begin {gather*} \left [\frac {3 \, {\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d - 45 \, a^{2} b c d^{2} + 35 \, a^{3} d^{3}\right )} \sqrt {a c} x^{6} e^{\frac {3}{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 - 100 \, a^{2} b c^{2} d^{2} + 105 \, a^{3} c d^{3}\right )} x^{6} + {\left (3 \, a b^{2} c^{4} - 38 \, a^{2} b c^{3} d + 35 \, a^{3} c^{2} d^{2}\right )} x^{4} + 14 \, {\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 {3}{2}}}{192 \, a^{2} c^{5} x^{6}}, -\frac {3 \, {\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d - 45 \, a^{2} b c d^{2} + 35 \, 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 {3}{2}} + 2 \, {\left (8 \, a^{3} c^{4} + {\left (3 \, a b^{2} c^{3} d - 100 \, a^{2} b c^{2} d^{2} + 105 \, a^{3} c d^{3}\right )} x^{6} + {\left (3 \, a b^{2} c^{4} - 38 \, a^{2} b c^{3} d + 35 \, a^{3} c^{2} d^{2}\right )} x^{4} + 14 \, {\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 {3}{2}}}{96 \, a^{2} c^{5} 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))^(3/2)/x^7,x, algorithm="fricas")

[Out]

[1/192*(3*(b^3*c^3 + 9*a*b^2*c^2*d - 45*a^2*b*c*d^2 + 35*a^3*d^3)*sqrt(a*c)*x^6*e^(3/2)*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 - 100*a^2*b*c^2*d^2 + 1
05*a^3*c*d^3)*x^6 + (3*a*b^2*c^4 - 38*a^2*b*c^3*d + 35*a^3*c^2*d^2)*x^4 + 14*(a^2*b*c^4 - a^3*c^3*d)*x^2)*sqrt
((b*x^2 + a)/(d*x^2 + c))*e^(3/2))/(a^2*c^5*x^6), -1/96*(3*(b^3*c^3 + 9*a*b^2*c^2*d - 45*a^2*b*c*d^2 + 35*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^(3/2) + 2*(8*a^3*c^4 + (3*a*b^2*c^3*d - 100*a^2*b*c^2*d^2 + 105*a^3*c*d^3)*x^6 + (3*a*b^2*c^4 - 38*
a^2*b*c^3*d + 35*a^3*c^2*d^2)*x^4 + 14*(a^2*b*c^4 - a^3*c^3*d)*x^2)*sqrt((b*x^2 + a)/(d*x^2 + c))*e^(3/2))/(a^
2*c^5*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))**(3/2)/x**7,x)

[Out]

Timed out

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*(b*x^2+a)/(d*x^2+c))^(3/2)/x^7,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}\right )}^{3/2}}{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))^(3/2)/x^7,x)

[Out]

int(((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^7, x)

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