\(\int \frac {x^4}{(a+b x^2) \sqrt {c+d x^2}} \, dx\) [708]

Optimal result

Integrand size = 24, antiderivative size = 114 \[ \int \frac {x^4}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {x \sqrt {c+d x^2}}{2 b d}+\frac {a^{3/2} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^2 \sqrt {b c-a d}}-\frac {(b c+2 a d) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 b^2 d^{3/2}} \]

[Out]

-1/2*(2*a*d+b*c)*arctanh(x*d^(1/2)/(d*x^2+c)^(1/2))/b^2/d^(3/2)+a^(3/2)*arctan(x*(-a*d+b*c)^(1/2)/a^(1/2)/(d*x
^2+c)^(1/2))/b^2/(-a*d+b*c)^(1/2)+1/2*x*(d*x^2+c)^(1/2)/b/d

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {490, 537, 223, 212, 385, 211} \[ \int \frac {x^4}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {a^{3/2} \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^2 \sqrt {b c-a d}}-\frac {(2 a d+b c) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 b^2 d^{3/2}}+\frac {x \sqrt {c+d x^2}}{2 b d} \]

[In]

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

[Out]

(x*Sqrt[c + d*x^2])/(2*b*d) + (a^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(b^2*Sqrt[b*c -
a*d]) - ((b*c + 2*a*d)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(2*b^2*d^(3/2))

Rule 211

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 385

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

Rule 490

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

Rule 537

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {c+d x^2}}{2 b d}-\frac {\int \frac {a c+(b c+2 a d) x^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 b d} \\ & = \frac {x \sqrt {c+d x^2}}{2 b d}+\frac {a^2 \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{b^2}-\frac {(b c+2 a d) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{2 b^2 d} \\ & = \frac {x \sqrt {c+d x^2}}{2 b d}+\frac {a^2 \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{b^2}-\frac {(b c+2 a d) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 b^2 d} \\ & = \frac {x \sqrt {c+d x^2}}{2 b d}+\frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^2 \sqrt {b c-a d}}-\frac {(b c+2 a d) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 b^2 d^{3/2}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(375\) vs. \(2(114)=228\).

Time = 2.14 (sec) , antiderivative size = 375, normalized size of antiderivative = 3.29 \[ \int \frac {x^4}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {x \sqrt {c+d x^2}}{2 b d}-\frac {\sqrt {a} \left (\sqrt {b} \sqrt {c}+\sqrt {b c-a d}\right ) \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{b^2 d \sqrt {b c-a d}}-\frac {\sqrt {a} \left (-\sqrt {b} \sqrt {c}+\sqrt {b c-a d}\right ) \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{b^2 d \sqrt {b c-a d}}+\frac {(-b c-2 a d) \text {arctanh}\left (\frac {\sqrt {d} x}{-\sqrt {c}+\sqrt {c+d x^2}}\right )}{b^2 d^{3/2}} \]

[In]

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

[Out]

(x*Sqrt[c + d*x^2])/(2*b*d) - (Sqrt[a]*(Sqrt[b]*Sqrt[c] + Sqrt[b*c - a*d])*Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c
]*Sqrt[b*c - a*d]]*ArcTan[(Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(-Sqrt[c] + Sqrt[
c + d*x^2]))])/(b^2*d*Sqrt[b*c - a*d]) - (Sqrt[a]*(-(Sqrt[b]*Sqrt[c]) + Sqrt[b*c - a*d])*Sqrt[2*b*c - a*d + 2*
Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*ArcTan[(Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(-S
qrt[c] + Sqrt[c + d*x^2]))])/(b^2*d*Sqrt[b*c - a*d]) + ((-(b*c) - 2*a*d)*ArcTanh[(Sqrt[d]*x)/(-Sqrt[c] + Sqrt[
c + d*x^2])])/(b^2*d^(3/2))

Maple [A] (verified)

Time = 3.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.98

method	result	size
pseudoelliptic	\(\frac {a^{2} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right ) d^{\frac {3}{2}}-\sqrt {\left (a d -b c \right ) a}\, \left (\left (\frac {b c}{2}+a d \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right )-\frac {\sqrt {d \,x^{2}+c}\, b x \sqrt {d}}{2}\right )}{\sqrt {\left (a d -b c \right ) a}\, d^{\frac {3}{2}} b^{2}}\)	\(112\)
risch	\(\frac {x \sqrt {d \,x^{2}+c}}{2 b d}-\frac {\frac {\left (2 a d +b c \right ) \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{b \sqrt {d}}-\frac {a^{2} d \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}+\frac {a^{2} d \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}}{2 b d}\)	\(377\)
default	\(\frac {\frac {x \sqrt {d \,x^{2}+c}}{2 d}-\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 d^{\frac {3}{2}}}}{b}-\frac {a \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{b^{2} \sqrt {d}}-\frac {a^{2} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 b^{2} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {a^{2} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 b^{2} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\)	\(385\)

[In]

int(x^4/(b*x^2+a)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

(a^2*arctanh((d*x^2+c)^(1/2)/x*a/((a*d-b*c)*a)^(1/2))*d^(3/2)-((a*d-b*c)*a)^(1/2)*((1/2*b*c+a*d)*arctanh((d*x^
2+c)^(1/2)/x/d^(1/2))-1/2*(d*x^2+c)^(1/2)*b*x*d^(1/2)))/((a*d-b*c)*a)^(1/2)/d^(3/2)/b^2

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 717, normalized size of antiderivative = 6.29 \[ \int \frac {x^4}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\left [\frac {a d^{2} \sqrt {-\frac {a}{b c - a d}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{3} - {\left (a b c^{2} - a^{2} c d\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 2 \, \sqrt {d x^{2} + c} b d x + {\left (b c + 2 \, a d\right )} \sqrt {d} \log \left (-2 \, d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right )}{4 \, b^{2} d^{2}}, \frac {a d^{2} \sqrt {-\frac {a}{b c - a d}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{3} - {\left (a b c^{2} - a^{2} c d\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 2 \, \sqrt {d x^{2} + c} b d x + 2 \, {\left (b c + 2 \, a d\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right )}{4 \, b^{2} d^{2}}, -\frac {2 \, a d^{2} \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{3} + a c x\right )}}\right ) - 2 \, \sqrt {d x^{2} + c} b d x - {\left (b c + 2 \, a d\right )} \sqrt {d} \log \left (-2 \, d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right )}{4 \, b^{2} d^{2}}, -\frac {a d^{2} \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{3} + a c x\right )}}\right ) - \sqrt {d x^{2} + c} b d x - {\left (b c + 2 \, a d\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right )}{2 \, b^{2} d^{2}}\right ] \]

[In]

integrate(x^4/(b*x^2+a)/(d*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

[1/4*(a*d^2*sqrt(-a/(b*c - a*d))*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c
*d)*x^2 + 4*((b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x^3 - (a*b*c^2 - a^2*c*d)*x)*sqrt(d*x^2 + c)*sqrt(-a/(b*c - a*d
)))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 2*sqrt(d*x^2 + c)*b*d*x + (b*c + 2*a*d)*sqrt(d)*log(-2*d*x^2 + 2*sqrt(d*x^2
+ c)*sqrt(d)*x - c))/(b^2*d^2), 1/4*(a*d^2*sqrt(-a/(b*c - a*d))*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 +
a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 + 4*((b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x^3 - (a*b*c^2 - a^2*c*d)*x)*sq
rt(d*x^2 + c)*sqrt(-a/(b*c - a*d)))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 2*sqrt(d*x^2 + c)*b*d*x + 2*(b*c + 2*a*d)*s
qrt(-d)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)))/(b^2*d^2), -1/4*(2*a*d^2*sqrt(a/(b*c - a*d))*arctan(-1/2*((b*c - 2
*a*d)*x^2 - a*c)*sqrt(d*x^2 + c)*sqrt(a/(b*c - a*d))/(a*d*x^3 + a*c*x)) - 2*sqrt(d*x^2 + c)*b*d*x - (b*c + 2*a
*d)*sqrt(d)*log(-2*d*x^2 + 2*sqrt(d*x^2 + c)*sqrt(d)*x - c))/(b^2*d^2), -1/2*(a*d^2*sqrt(a/(b*c - a*d))*arctan
(-1/2*((b*c - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c)*sqrt(a/(b*c - a*d))/(a*d*x^3 + a*c*x)) - sqrt(d*x^2 + c)*b*d*x
- (b*c + 2*a*d)*sqrt(-d)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)))/(b^2*d^2)]

Sympy [F]

\[ \int \frac {x^4}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\int \frac {x^{4}}{\left (a + b x^{2}\right ) \sqrt {c + d x^{2}}}\, dx \]

[In]

integrate(x**4/(b*x**2+a)/(d*x**2+c)**(1/2),x)

[Out]

Integral(x**4/((a + b*x**2)*sqrt(c + d*x**2)), x)

Maxima [F]

\[ \int \frac {x^4}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\int { \frac {x^{4}}{{\left (b x^{2} + a\right )} \sqrt {d x^{2} + c}} \,d x } \]

[In]

integrate(x^4/(b*x^2+a)/(d*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {x^4}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^4/(b*x^2+a)/(d*x^2+c)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m i_lex_is_greater Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\int \frac {x^4}{\left (b\,x^2+a\right )\,\sqrt {d\,x^2+c}} \,d x \]

[In]

int(x^4/((a + b*x^2)*(c + d*x^2)^(1/2)),x)

[Out]

int(x^4/((a + b*x^2)*(c + d*x^2)^(1/2)), x)