3.4.37 \(\int \frac {(a+\frac {b}{c+d x^2})^{3/2}}{x^7} \, dx\) [337]

Optimal. Leaf size=292 \[ -\frac {b d^3 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{c^4}-\frac {\left (79 b^2+108 a b c+24 a^2 c^2\right ) d^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{48 c^4 (b+a c) x^2}+\frac {(11 b+12 a c) d \left (c+d x^2\right )^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{24 c^4 x^4}-\frac {\left (c+d x^2\right )^3 \left (\frac {b+a c+a d x^2}{c+d x^2}\right )^{5/2}}{6 c^2 (b+a c) x^6}+\frac {b \left (35 b^2+60 a b c+24 a^2 c^2\right ) d^3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {b+a c}}\right )}{16 c^{9/2} (b+a c)^{3/2}} \]

[Out]

-1/6*(d*x^2+c)^3*((a*d*x^2+a*c+b)/(d*x^2+c))^(5/2)/c^2/(a*c+b)/x^6+1/16*b*(24*a^2*c^2+60*a*b*c+35*b^2)*d^3*arc
tanh(c^(1/2)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/(a*c+b)^(1/2))/c^(9/2)/(a*c+b)^(3/2)-b*d^3*((a*d*x^2+a*c+b)/(d*
x^2+c))^(1/2)/c^4-1/48*(24*a^2*c^2+108*a*b*c+79*b^2)*d^2*(d*x^2+c)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/c^4/(a*c+
b)/x^2+1/24*(12*a*c+11*b)*d*(d*x^2+c)^2*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/c^4/x^4

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Rubi [A]
time = 0.42, antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1985, 1981, 1980, 474, 466, 1171, 396, 214} \begin {gather*} \frac {b d^3 \left (24 a^2 c^2+60 a b c+35 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a c+b}}\right )}{16 c^{9/2} (a c+b)^{3/2}}-\frac {d^2 \left (24 a^2 c^2+108 a b c+79 b^2\right ) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{48 c^4 x^2 (a c+b)}-\frac {b d^3 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{c^4}+\frac {d (12 a c+11 b) \left (c+d x^2\right )^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{24 c^4 x^4}-\frac {\left (c+d x^2\right )^3 \left (\frac {a c+a d x^2+b}{c+d x^2}\right )^{5/2}}{6 c^2 x^6 (a c+b)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b/(c + d*x^2))^(3/2)/x^7,x]

[Out]

-((b*d^3*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/c^4) - ((79*b^2 + 108*a*b*c + 24*a^2*c^2)*d^2*(c + d*x^2)*Sqrt
[(b + a*c + a*d*x^2)/(c + d*x^2)])/(48*c^4*(b + a*c)*x^2) + ((11*b + 12*a*c)*d*(c + d*x^2)^2*Sqrt[(b + a*c + a
*d*x^2)/(c + d*x^2)])/(24*c^4*x^4) - ((c + d*x^2)^3*((b + a*c + a*d*x^2)/(c + d*x^2))^(5/2))/(6*c^2*(b + a*c)*
x^6) + (b*(35*b^2 + 60*a*b*c + 24*a^2*c^2)*d^3*ArcTanh[(Sqrt[c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/Sqrt[b
+ a*c]])/(16*c^(9/2)*(b + a*c)^(3/2))

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]]

Rule 1985

Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u*((b + a*c + a*d*x^n)/(c + d*x^n))^p
, x] /; FreeQ[{a, b, c, d, n, p}, x]

Rubi steps

\begin {align*} \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^7} \, dx &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {\left (b+a \left (c+d x^2\right )\right )^{3/2}}{x^7 \left (c+d x^2\right )^{3/2}} \, dx}{\sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {\left (b+a c+a d x^2\right )^{3/2}}{x^7 \left (c+d x^2\right )^{3/2}} \, dx}{\sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \text {Subst}\left (\int \frac {(b+a c+a d x)^{3/2}}{x^4 (c+d x)^{3/2}} \, dx,x,x^2\right )}{2 \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {(b+a c) \sqrt {a+\frac {b}{c+d x^2}}}{6 c x^6}-\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \text {Subst}\left (\int \frac {\frac {7}{2} b (b+a c) d+3 a b d^2 x}{x^3 (c+d x)^{3/2} \sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{6 c \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {(b+a c) \sqrt {a+\frac {b}{c+d x^2}}}{6 c x^6}+\frac {7 b d \sqrt {a+\frac {b}{c+d x^2}}}{24 c^2 x^4}+\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \text {Subst}\left (\int \frac {\frac {1}{4} b (b+a c) (35 b+32 a c) d^2+7 a b (b+a c) d^3 x}{x^2 (c+d x)^{3/2} \sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{12 c^2 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {(b+a c) \sqrt {a+\frac {b}{c+d x^2}}}{6 c x^6}+\frac {7 b d \sqrt {a+\frac {b}{c+d x^2}}}{24 c^2 x^4}-\frac {b (35 b+32 a c) d^2 \sqrt {a+\frac {b}{c+d x^2}}}{48 c^3 (b+a c) x^2}-\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \text {Subst}\left (\int \frac {\frac {3}{8} b (b+a c) \left (35 b^2+60 a b c+24 a^2 c^2\right ) d^3+\frac {1}{4} a b (b+a c) (35 b+32 a c) d^4 x}{x (c+d x)^{3/2} \sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{12 c^3 (b+a c)^2 \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (105 b^2+110 a b c+8 a^2 c^2\right ) d^3 \sqrt {a+\frac {b}{c+d x^2}}}{48 c^4 (b+a c)}-\frac {(b+a c) \sqrt {a+\frac {b}{c+d x^2}}}{6 c x^6}+\frac {7 b d \sqrt {a+\frac {b}{c+d x^2}}}{24 c^2 x^4}-\frac {b (35 b+32 a c) d^2 \sqrt {a+\frac {b}{c+d x^2}}}{48 c^3 (b+a c) x^2}-\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \text {Subst}\left (\int \frac {3 b^2 (b+a c) \left (35 b^2+60 a b c+24 a^2 c^2\right ) d^4}{16 x \sqrt {c+d x} \sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{6 b c^4 (b+a c)^2 d \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (105 b^2+110 a b c+8 a^2 c^2\right ) d^3 \sqrt {a+\frac {b}{c+d x^2}}}{48 c^4 (b+a c)}-\frac {(b+a c) \sqrt {a+\frac {b}{c+d x^2}}}{6 c x^6}+\frac {7 b d \sqrt {a+\frac {b}{c+d x^2}}}{24 c^2 x^4}-\frac {b (35 b+32 a c) d^2 \sqrt {a+\frac {b}{c+d x^2}}}{48 c^3 (b+a c) x^2}-\frac {\left (b \left (35 b^2+60 a b c+24 a^2 c^2\right ) d^3 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x} \sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{32 c^4 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (105 b^2+110 a b c+8 a^2 c^2\right ) d^3 \sqrt {a+\frac {b}{c+d x^2}}}{48 c^4 (b+a c)}-\frac {(b+a c) \sqrt {a+\frac {b}{c+d x^2}}}{6 c x^6}+\frac {7 b d \sqrt {a+\frac {b}{c+d x^2}}}{24 c^2 x^4}-\frac {b (35 b+32 a c) d^2 \sqrt {a+\frac {b}{c+d x^2}}}{48 c^3 (b+a c) x^2}-\frac {\left (b \left (35 b^2+60 a b c+24 a^2 c^2\right ) d^3 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \text {Subst}\left (\int \frac {1}{-c-(-b-a c) x^2} \, dx,x,\frac {\sqrt {c+d x^2}}{\sqrt {b+a \left (c+d x^2\right )}}\right )}{16 c^4 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (105 b^2+110 a b c+8 a^2 c^2\right ) d^3 \sqrt {a+\frac {b}{c+d x^2}}}{48 c^4 (b+a c)}-\frac {(b+a c) \sqrt {a+\frac {b}{c+d x^2}}}{6 c x^6}+\frac {7 b d \sqrt {a+\frac {b}{c+d x^2}}}{24 c^2 x^4}-\frac {b (35 b+32 a c) d^2 \sqrt {a+\frac {b}{c+d x^2}}}{48 c^3 (b+a c) x^2}+\frac {b \left (35 b^2+60 a b c+24 a^2 c^2\right ) d^3 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}} \tanh ^{-1}\left (\frac {\sqrt {b+a c} \sqrt {c+d x^2}}{\sqrt {c} \sqrt {b+a \left (c+d x^2\right )}}\right )}{16 c^{9/2} (b+a c)^{3/2} \sqrt {b+a \left (c+d x^2\right )}}\\ \end {align*}

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Mathematica [A]
time = 0.45, size = 226, normalized size = 0.77 \begin {gather*} \frac {-\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (8 a^2 c^2 \left (c^3+d^3 x^6\right )+2 a b c \left (8 c^3-7 c^2 d x^2+16 c d^2 x^4+55 d^3 x^6\right )+b^2 \left (8 c^3-14 c^2 d x^2+35 c d^2 x^4+105 d^3 x^6\right )\right )}{(b+a c) x^6}+\frac {3 b \left (35 b^2+60 a b c+24 a^2 c^2\right ) d^3 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {-b-a c}}\right )}{(-b-a c)^{3/2}}}{48 c^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-((Sqrt[c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(8*a^2*c^2*(c^3 + d^3*x^6) + 2*a*b*c*(8*c^3 - 7*c^2*d*x^2 +
16*c*d^2*x^4 + 55*d^3*x^6) + b^2*(8*c^3 - 14*c^2*d*x^2 + 35*c*d^2*x^4 + 105*d^3*x^6)))/((b + a*c)*x^6)) + (3*b
*(35*b^2 + 60*a*b*c + 24*a^2*c^2)*d^3*ArcTan[(Sqrt[c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/Sqrt[-b - a*c]])/
(-b - a*c)^(3/2))/(48*c^(9/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2604\) vs. \(2(266)=532\).
time = 0.15, size = 2605, normalized size = 8.92

method	result	size
risch	\(-\frac {\left (d \,x^{2}+c \right ) \left (8 a^{2} c^{2} d^{2} x^{4}+62 a c \,d^{2} b \,x^{4}-8 a^{2} c^{3} d \,x^{2}+57 b^{2} d^{2} x^{4}-30 a b \,c^{2} d \,x^{2}+8 a^{2} c^{4}-22 b^{2} c d \,x^{2}+16 a b \,c^{3}+8 b^{2} c^{2}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{48 c^{4} x^{6} \left (a c +b \right )}+\frac {\left (\frac {3 d^{3} b \ln \left (\frac {2 c^{2} a +2 b c +\left (2 a c d +b d \right ) x^{2}+2 \sqrt {c^{2} a +b c}\, \sqrt {c^{2} a +b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}}{x^{2}}\right ) a^{2}}{4 c^{2} \left (a c +b \right ) \sqrt {c^{2} a +b c}}+\frac {15 d^{3} b^{2} \ln \left (\frac {2 c^{2} a +2 b c +\left (2 a c d +b d \right ) x^{2}+2 \sqrt {c^{2} a +b c}\, \sqrt {c^{2} a +b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}}{x^{2}}\right ) a}{8 c^{3} \left (a c +b \right ) \sqrt {c^{2} a +b c}}+\frac {35 d^{3} b^{3} \ln \left (\frac {2 c^{2} a +2 b c +\left (2 a c d +b d \right ) x^{2}+2 \sqrt {c^{2} a +b c}\, \sqrt {c^{2} a +b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}}{x^{2}}\right )}{32 c^{4} \left (a c +b \right ) \sqrt {c^{2} a +b c}}-\frac {d^{4} b \,x^{2} a^{2}}{c^{3} \left (a c +b \right ) \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}-\frac {d^{4} b^{2} x^{2} a}{c^{4} \left (a c +b \right ) \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}-\frac {d^{3} b \,a^{2}}{c^{2} \left (a c +b \right ) \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}-\frac {2 d^{3} b^{2} a}{c^{3} \left (a c +b \right ) \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}-\frac {d^{3} b^{3}}{c^{4} \left (a c +b \right ) \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}{a d \,x^{2}+a c +b}\)	\(775\)
default	\(\text {Expression too large to display}\)	\(2605\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)*(-105*ln((2*a*c*d*x^2+b*d*x^2+2*c^2*a+2*(a*c^2+b*c)^(1/2)*(a*d^2*x^4+2
*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)+2*b*c)/x^2)*b^6*c^3*d^4*x^8-174*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^
(1/2)*(a*c^2+b*c)^(5/2)*b^3*d^4*x^8+16*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(3/2)*(a*c^2+b*c)^(5/2)*a^2*c
^5+16*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(3/2)*(a*c^2+b*c)^(5/2)*b^2*c^3-927*ln((2*a*c*d*x^2+b*d*x^2+2*
c^2*a+2*(a*c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)+2*b*c)/x^2)*a^2*b^4*c^6*d^3*x^6+96*(
a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(3/2)*(a*c^2+b*c)^(5/2)*a^2*c^2*d^3*x^6-495*ln((2*a*c*d*x^2+b*d*x^2+2
*c^2*a+2*(a*c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)+2*b*c)/x^2)*a*b^5*c^5*d^3*x^6+48*(a
*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(3/2)*(a*c^2+b*c)^(5/2)*a^2*c^3*d^2*x^4-174*(a*d^2*x^4+2*a*c*d*x^2+b*d
*x^2+a*c^2+b*c)^(1/2)*(a*c^2+b*c)^(5/2)*b^3*c*d^3*x^6+96*(a*c^2+b*c)^(5/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*b
^3*c*d^3*x^6-32*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(3/2)*(a*c^2+b*c)^(5/2)*a^2*c^4*d*x^2+114*(a*d^2*x^4
+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(3/2)*(a*c^2+b*c)^(5/2)*b^2*c*d^2*x^4-44*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+
b*c)^(3/2)*(a*c^2+b*c)^(5/2)*b^2*c^2*d*x^2+276*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(3/2)*(a*c^2+b*c)^(5/
2)*a*b*c*d^3*x^6-564*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*c^2+b*c)^(5/2)*a*b^2*c^2*d^3*x^6+192*(
a*c^2+b*c)^(5/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*a*b^2*c^2*d^3*x^6+168*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+
b*c)^(3/2)*(a*c^2+b*c)^(5/2)*a*b*c^2*d^2*x^4-76*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(3/2)*(a*c^2+b*c)^(5
/2)*a*b*c^3*d*x^2-105*ln((2*a*c*d*x^2+b*d*x^2+2*c^2*a+2*(a*c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2
+b*c)^(1/2)+2*b*c)/x^2)*b^6*c^4*d^3*x^6+174*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(3/2)*(a*c^2+b*c)^(5/2)*
b^2*d^3*x^6-276*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*c^2+b*c)^(5/2)*a^2*b*c*d^5*x^10-816*(a*d^2*
x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*c^2+b*c)^(5/2)*a^2*b*c^2*d^4*x^8-540*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^
2+a*c^2+b*c)^(1/2)*(a*c^2+b*c)^(5/2)*a^2*b*c^3*d^3*x^6+96*(a*c^2+b*c)^(5/2)*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*
a^2*b*c^3*d^3*x^6-738*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*c^2+b*c)^(5/2)*a*b^2*c*d^4*x^8+32*(a*
d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(3/2)*(a*c^2+b*c)^(5/2)*a*b*c^4-72*ln((2*a*c*d*x^2+b*d*x^2+2*c^2*a+2*(a
*c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)+2*b*c)/x^2)*a^5*b*c^8*d^4*x^8-96*(a*d^2*x^4+2*
a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*c^2+b*c)^(5/2)*a^3*c^2*d^5*x^10-396*ln((2*a*c*d*x^2+b*d*x^2+2*c^2*a+2*(a
*c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)+2*b*c)/x^2)*a^4*b^2*c^7*d^4*x^8-72*ln((2*a*c*d
*x^2+b*d*x^2+2*c^2*a+2*(a*c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)+2*b*c)/x^2)*a^5*b*c^9
*d^3*x^6-861*ln((2*a*c*d*x^2+b*d*x^2+2*c^2*a+2*(a*c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/
2)+2*b*c)/x^2)*a^3*b^3*c^6*d^4*x^8-240*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*c^2+b*c)^(5/2)*a^3*c
^3*d^4*x^8-174*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*c^2+b*c)^(5/2)*a*b^2*d^5*x^10-396*ln((2*a*c*
d*x^2+b*d*x^2+2*c^2*a+2*(a*c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)+2*b*c)/x^2)*a^4*b^2*
c^8*d^3*x^6-927*ln((2*a*c*d*x^2+b*d*x^2+2*c^2*a+2*(a*c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^
(1/2)+2*b*c)/x^2)*a^2*b^4*c^5*d^4*x^8-861*ln((2*a*c*d*x^2+b*d*x^2+2*c^2*a+2*(a*c^2+b*c)^(1/2)*(a*d^2*x^4+2*a*c
*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)+2*b*c)/x^2)*a^3*b^3*c^7*d^3*x^6-495*ln((2*a*c*d*x^2+b*d*x^2+2*c^2*a+2*(a*c^2+b
*c)^(1/2)*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)+2*b*c)/x^2)*a*b^5*c^4*d^4*x^8-144*(a*d^2*x^4+2*a*c*d
*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*c^2+b*c)^(5/2)*a^3*c^4*d^3*x^6)/(a*c^2+b*c)^(5/2)/x^6/(a*c+b)^2/c^5/((d*x^2+c
)*(a*d*x^2+a*c+b))^(1/2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 534 vs. \(2 (266) = 532\).
time = 0.54, size = 534, normalized size = 1.83 \begin {gather*} -\frac {{\left (24 \, a^{2} b c^{2} + 60 \, a b^{2} c + 35 \, b^{3}\right )} d^{3} \log \left (\frac {c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} - \sqrt {{\left (a c + b\right )} c}}{c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} + \sqrt {{\left (a c + b\right )} c}}\right )}{32 \, {\left (a c^{5} + b c^{4}\right )} \sqrt {{\left (a c + b\right )} c}} - \frac {b d^{3} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{c^{4}} - \frac {3 \, {\left (8 \, a^{2} b c^{4} + 36 \, a b^{2} c^{3} + 29 \, b^{3} c^{2}\right )} d^{3} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {5}{2}} - 8 \, {\left (6 \, a^{3} b c^{4} + 30 \, a^{2} b^{2} c^{3} + 41 \, a b^{3} c^{2} + 17 \, b^{4} c\right )} d^{3} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {3}{2}} + 3 \, {\left (8 \, a^{4} b c^{4} + 44 \, a^{3} b^{2} c^{3} + 83 \, a^{2} b^{3} c^{2} + 66 \, a b^{4} c + 19 \, b^{5}\right )} d^{3} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{48 \, {\left (a^{4} c^{8} + 4 \, a^{3} b c^{7} + 6 \, a^{2} b^{2} c^{6} + 4 \, a b^{3} c^{5} + b^{4} c^{4} - \frac {{\left (a c^{8} + b c^{7}\right )} {\left (a d x^{2} + a c + b\right )}^{3}}{{\left (d x^{2} + c\right )}^{3}} + \frac {3 \, {\left (a^{2} c^{8} + 2 \, a b c^{7} + b^{2} c^{6}\right )} {\left (a d x^{2} + a c + b\right )}^{2}}{{\left (d x^{2} + c\right )}^{2}} - \frac {3 \, {\left (a^{3} c^{8} + 3 \, a^{2} b c^{7} + 3 \, a b^{2} c^{6} + b^{3} c^{5}\right )} {\left (a d x^{2} + a c + b\right )}}{d x^{2} + c}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*(24*a^2*b*c^2 + 60*a*b^2*c + 35*b^3)*d^3*log((c*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)) - sqrt((a*c + b)*c
))/(c*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)) + sqrt((a*c + b)*c)))/((a*c^5 + b*c^4)*sqrt((a*c + b)*c)) - b*d^3*
sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))/c^4 - 1/48*(3*(8*a^2*b*c^4 + 36*a*b^2*c^3 + 29*b^3*c^2)*d^3*((a*d*x^2 +
a*c + b)/(d*x^2 + c))^(5/2) - 8*(6*a^3*b*c^4 + 30*a^2*b^2*c^3 + 41*a*b^3*c^2 + 17*b^4*c)*d^3*((a*d*x^2 + a*c +
b)/(d*x^2 + c))^(3/2) + 3*(8*a^4*b*c^4 + 44*a^3*b^2*c^3 + 83*a^2*b^3*c^2 + 66*a*b^4*c + 19*b^5)*d^3*sqrt((a*d
*x^2 + a*c + b)/(d*x^2 + c)))/(a^4*c^8 + 4*a^3*b*c^7 + 6*a^2*b^2*c^6 + 4*a*b^3*c^5 + b^4*c^4 - (a*c^8 + b*c^7)
*(a*d*x^2 + a*c + b)^3/(d*x^2 + c)^3 + 3*(a^2*c^8 + 2*a*b*c^7 + b^2*c^6)*(a*d*x^2 + a*c + b)^2/(d*x^2 + c)^2 -
3*(a^3*c^8 + 3*a^2*b*c^7 + 3*a*b^2*c^6 + b^3*c^5)*(a*d*x^2 + a*c + b)/(d*x^2 + c))

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Fricas [A]
time = 0.95, size = 733, normalized size = 2.51 \begin {gather*} \left [\frac {3 \, {\left (24 \, a^{2} b c^{2} + 60 \, a b^{2} c + 35 \, b^{3}\right )} \sqrt {a c^{2} + b c} d^{3} x^{6} \log \left (\frac {{\left (8 \, a^{2} c^{2} + 8 \, a b c + b^{2}\right )} d^{2} x^{4} + 8 \, a^{2} c^{4} + 16 \, a b c^{3} + 8 \, b^{2} c^{2} + 8 \, {\left (2 \, a^{2} c^{3} + 3 \, a b c^{2} + b^{2} c\right )} d x^{2} + 4 \, {\left ({\left (2 \, a c + b\right )} d^{2} x^{4} + 2 \, a c^{3} + {\left (4 \, a c^{2} + 3 \, b c\right )} d x^{2} + 2 \, b c^{2}\right )} \sqrt {a c^{2} + b c} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{x^{4}}\right ) - 4 \, {\left (8 \, a^{3} c^{7} + {\left (8 \, a^{3} c^{4} + 118 \, a^{2} b c^{3} + 215 \, a b^{2} c^{2} + 105 \, b^{3} c\right )} d^{3} x^{6} + 24 \, a^{2} b c^{6} + 24 \, a b^{2} c^{5} + 8 \, b^{3} c^{4} + {\left (32 \, a^{2} b c^{4} + 67 \, a b^{2} c^{3} + 35 \, b^{3} c^{2}\right )} d^{2} x^{4} - 14 \, {\left (a^{2} b c^{5} + 2 \, a b^{2} c^{4} + b^{3} c^{3}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{192 \, {\left (a^{2} c^{7} + 2 \, a b c^{6} + b^{2} c^{5}\right )} x^{6}}, -\frac {3 \, {\left (24 \, a^{2} b c^{2} + 60 \, a b^{2} c + 35 \, b^{3}\right )} \sqrt {-a c^{2} - b c} d^{3} x^{6} \arctan \left (\frac {{\left ({\left (2 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + 2 \, b c\right )} \sqrt {-a c^{2} - b c} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} c^{3} + 2 \, a b c^{2} + {\left (a^{2} c^{2} + a b c\right )} d x^{2} + b^{2} c\right )}}\right ) + 2 \, {\left (8 \, a^{3} c^{7} + {\left (8 \, a^{3} c^{4} + 118 \, a^{2} b c^{3} + 215 \, a b^{2} c^{2} + 105 \, b^{3} c\right )} d^{3} x^{6} + 24 \, a^{2} b c^{6} + 24 \, a b^{2} c^{5} + 8 \, b^{3} c^{4} + {\left (32 \, a^{2} b c^{4} + 67 \, a b^{2} c^{3} + 35 \, b^{3} c^{2}\right )} d^{2} x^{4} - 14 \, {\left (a^{2} b c^{5} + 2 \, a b^{2} c^{4} + b^{3} c^{3}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{96 \, {\left (a^{2} c^{7} + 2 \, a b c^{6} + b^{2} c^{5}\right )} x^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/192*(3*(24*a^2*b*c^2 + 60*a*b^2*c + 35*b^3)*sqrt(a*c^2 + b*c)*d^3*x^6*log(((8*a^2*c^2 + 8*a*b*c + b^2)*d^2*
x^4 + 8*a^2*c^4 + 16*a*b*c^3 + 8*b^2*c^2 + 8*(2*a^2*c^3 + 3*a*b*c^2 + b^2*c)*d*x^2 + 4*((2*a*c + b)*d^2*x^4 +
2*a*c^3 + (4*a*c^2 + 3*b*c)*d*x^2 + 2*b*c^2)*sqrt(a*c^2 + b*c)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/x^4) - 4
*(8*a^3*c^7 + (8*a^3*c^4 + 118*a^2*b*c^3 + 215*a*b^2*c^2 + 105*b^3*c)*d^3*x^6 + 24*a^2*b*c^6 + 24*a*b^2*c^5 +
8*b^3*c^4 + (32*a^2*b*c^4 + 67*a*b^2*c^3 + 35*b^3*c^2)*d^2*x^4 - 14*(a^2*b*c^5 + 2*a*b^2*c^4 + b^3*c^3)*d*x^2)
*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/((a^2*c^7 + 2*a*b*c^6 + b^2*c^5)*x^6), -1/96*(3*(24*a^2*b*c^2 + 60*a*b
^2*c + 35*b^3)*sqrt(-a*c^2 - b*c)*d^3*x^6*arctan(1/2*((2*a*c + b)*d*x^2 + 2*a*c^2 + 2*b*c)*sqrt(-a*c^2 - b*c)*
sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))/(a^2*c^3 + 2*a*b*c^2 + (a^2*c^2 + a*b*c)*d*x^2 + b^2*c)) + 2*(8*a^3*c^7
+ (8*a^3*c^4 + 118*a^2*b*c^3 + 215*a*b^2*c^2 + 105*b^3*c)*d^3*x^6 + 24*a^2*b*c^6 + 24*a*b^2*c^5 + 8*b^3*c^4 +
(32*a^2*b*c^4 + 67*a*b^2*c^3 + 35*b^3*c^2)*d^2*x^4 - 14*(a^2*b*c^5 + 2*a*b^2*c^4 + b^3*c^3)*d*x^2)*sqrt((a*d*x
^2 + a*c + b)/(d*x^2 + c)))/((a^2*c^7 + 2*a*b*c^6 + b^2*c^5)*x^6)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}}{x^{7}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/(d*x**2+c))**(3/2)/x**7,x)

[Out]

Integral(((a*c + a*d*x**2 + b)/(c + d*x**2))**(3/2)/x**7, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

undef

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

[Out]

int((a + b/(c + d*x^2))^(3/2)/x^7, x)

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