\(\int \frac {x^4 (a+b x^2)^2}{(c+d x^2)^{5/2}} \, dx\) [660]

Optimal result

Integrand size = 24, antiderivative size = 202 \[ \int \frac {x^4 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {(b c-a d)^2 x^5}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) x^3}{12 c d^3 \sqrt {c+d x^2}}+\frac {b^2 x^5}{4 d^2 \sqrt {c+d x^2}}-\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) x \sqrt {c+d x^2}}{8 c d^4}+\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 d^{9/2}} \]

[Out]

1/3*(-a*d+b*c)^2*x^5/c/d^2/(d*x^2+c)^(3/2)+1/8*(8*a^2*d^2-40*a*b*c*d+35*b^2*c^2)*arctanh(x*d^(1/2)/(d*x^2+c)^(
1/2))/d^(9/2)+1/12*(8*a^2*d^2-40*a*b*c*d+35*b^2*c^2)*x^3/c/d^3/(d*x^2+c)^(1/2)+1/4*b^2*x^5/d^2/(d*x^2+c)^(1/2)
-1/8*(8*a^2*d^2-40*a*b*c*d+35*b^2*c^2)*x*(d*x^2+c)^(1/2)/c/d^4

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 202, 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 = {474, 470, 294, 327, 223, 212} \[ \int \frac {x^4 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {\left (8 a^2 d^2-40 a b c d+35 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 d^{9/2}}-\frac {x \sqrt {c+d x^2} \left (8 a^2 d^2-40 a b c d+35 b^2 c^2\right )}{8 c d^4}+\frac {x^3 \left (8 a^2 d^2-40 a b c d+35 b^2 c^2\right )}{12 c d^3 \sqrt {c+d x^2}}+\frac {x^5 (b c-a d)^2}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {b^2 x^5}{4 d^2 \sqrt {c+d x^2}} \]

[In]

Int[(x^4*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]

[Out]

((b*c - a*d)^2*x^5)/(3*c*d^2*(c + d*x^2)^(3/2)) + ((35*b^2*c^2 - 40*a*b*c*d + 8*a^2*d^2)*x^3)/(12*c*d^3*Sqrt[c
+ d*x^2]) + (b^2*x^5)/(4*d^2*Sqrt[c + d*x^2]) - ((35*b^2*c^2 - 40*a*b*c*d + 8*a^2*d^2)*x*Sqrt[c + d*x^2])/(8*
c*d^4) + ((35*b^2*c^2 - 40*a*b*c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*d^(9/2))

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 294

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

Rule 327

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

Rule 470

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

Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d)^2 x^5}{3 c d^2 \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {x^4 \left (-3 a^2 d^2+5 (b c-a d)^2-3 b^2 c d x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c d^2} \\ & = \frac {(b c-a d)^2 x^5}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {b^2 x^5}{4 d^2 \sqrt {c+d x^2}}-\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) \int \frac {x^4}{\left (c+d x^2\right )^{3/2}} \, dx}{12 c d^2} \\ & = \frac {(b c-a d)^2 x^5}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) x^3}{12 c d^3 \sqrt {c+d x^2}}+\frac {b^2 x^5}{4 d^2 \sqrt {c+d x^2}}-\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) \int \frac {x^2}{\sqrt {c+d x^2}} \, dx}{4 c d^3} \\ & = \frac {(b c-a d)^2 x^5}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) x^3}{12 c d^3 \sqrt {c+d x^2}}+\frac {b^2 x^5}{4 d^2 \sqrt {c+d x^2}}-\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) x \sqrt {c+d x^2}}{8 c d^4}+\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{8 d^4} \\ & = \frac {(b c-a d)^2 x^5}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) x^3}{12 c d^3 \sqrt {c+d x^2}}+\frac {b^2 x^5}{4 d^2 \sqrt {c+d x^2}}-\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) x \sqrt {c+d x^2}}{8 c d^4}+\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{8 d^4} \\ & = \frac {(b c-a d)^2 x^5}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) x^3}{12 c d^3 \sqrt {c+d x^2}}+\frac {b^2 x^5}{4 d^2 \sqrt {c+d x^2}}-\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) x \sqrt {c+d x^2}}{8 c d^4}+\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 d^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.81 \[ \int \frac {x^4 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {x \left (-8 a^2 d^2 \left (3 c+4 d x^2\right )+8 a b d \left (15 c^2+20 c d x^2+3 d^2 x^4\right )-b^2 \left (105 c^3+140 c^2 d x^2+21 c d^2 x^4-6 d^3 x^6\right )\right )}{24 d^4 \left (c+d x^2\right )^{3/2}}+\frac {\left (35 b^2 c^2-40 a b c d+8 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{-\sqrt {c}+\sqrt {c+d x^2}}\right )}{4 d^{9/2}} \]

[In]

Integrate[(x^4*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]

[Out]

(x*(-8*a^2*d^2*(3*c + 4*d*x^2) + 8*a*b*d*(15*c^2 + 20*c*d*x^2 + 3*d^2*x^4) - b^2*(105*c^3 + 140*c^2*d*x^2 + 21
*c*d^2*x^4 - 6*d^3*x^6)))/(24*d^4*(c + d*x^2)^(3/2)) + ((35*b^2*c^2 - 40*a*b*c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]
*x)/(-Sqrt[c] + Sqrt[c + d*x^2])])/(4*d^(9/2))

Maple [A] (verified)

Time = 3.03 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.73

method	result	size
pseudoelliptic	\(\frac {5 x b \left (-\frac {7 b \,x^{2}}{6}+a \right ) c^{2} d^{\frac {3}{2}}-x \left (\frac {7}{8} b^{2} x^{4}-\frac {20}{3} a b \,x^{2}+a^{2}\right ) c \,d^{\frac {5}{2}}+\left (\frac {1}{4} b^{2} x^{7}+a b \,x^{5}-\frac {4}{3} a^{2} x^{3}\right ) d^{\frac {7}{2}}-\frac {35 \sqrt {d}\, b^{2} c^{3} x}{8}+\left (a^{2} d^{2}-5 a b c d +\frac {35}{8} b^{2} c^{2}\right ) \left (d \,x^{2}+c \right )^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right )}{\left (d \,x^{2}+c \right )^{\frac {3}{2}} d^{\frac {9}{2}}}\)	\(147\)
default	\(b^{2} \left (\frac {x^{7}}{4 d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {7 c \left (\frac {x^{5}}{2 d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {5 c \left (-\frac {x^{3}}{3 d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {-\frac {x}{d \sqrt {d \,x^{2}+c}}+\frac {\ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{d^{\frac {3}{2}}}}{d}\right )}{2 d}\right )}{4 d}\right )+a^{2} \left (-\frac {x^{3}}{3 d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {-\frac {x}{d \sqrt {d \,x^{2}+c}}+\frac {\ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{d^{\frac {3}{2}}}}{d}\right )+2 a b \left (\frac {x^{5}}{2 d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {5 c \left (-\frac {x^{3}}{3 d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {-\frac {x}{d \sqrt {d \,x^{2}+c}}+\frac {\ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{d^{\frac {3}{2}}}}{d}\right )}{2 d}\right )\)	\(260\)
risch	\(\frac {b x \left (2 b d \,x^{2}+8 a d -11 b c \right ) \sqrt {d \,x^{2}+c}}{8 d^{4}}+\frac {8 a^{2} d^{\frac {3}{2}} \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )+\frac {35 b^{2} c^{2} \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{\sqrt {d}}-40 a b c \sqrt {d}\, \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )-\frac {2 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {\sqrt {d \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}-2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}}{3 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}}-\frac {\sqrt {d \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}-2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}}{3 c \left (x +\frac {\sqrt {-c d}}{d}\right )}\right )}{d}-\frac {2 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (-\frac {\sqrt {d \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}+2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}}{3 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}}-\frac {\sqrt {d \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}+2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}}{3 c \left (x -\frac {\sqrt {-c d}}{d}\right )}\right )}{d}-\frac {2 \left (3 a^{2} d^{2}-10 a b c d +7 b^{2} c^{2}\right ) \sqrt {d \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}-2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}}{d \left (x +\frac {\sqrt {-c d}}{d}\right )}-\frac {2 \left (3 a^{2} d^{2}-10 a b c d +7 b^{2} c^{2}\right ) \sqrt {d \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}+2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}}{d \left (x -\frac {\sqrt {-c d}}{d}\right )}}{8 d^{4}}\)	\(580\)

[In]

int(x^4*(b*x^2+a)^2/(d*x^2+c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/(d*x^2+c)^(3/2)/d^(9/2)*(5*x*b*(-7/6*b*x^2+a)*c^2*d^(3/2)-x*(7/8*b^2*x^4-20/3*a*b*x^2+a^2)*c*d^(5/2)+(1/4*b^
2*x^7+a*b*x^5-4/3*a^2*x^3)*d^(7/2)-35/8*d^(1/2)*b^2*c^3*x+(a^2*d^2-5*a*b*c*d+35/8*b^2*c^2)*(d*x^2+c)^(3/2)*arc
tanh((d*x^2+c)^(1/2)/x/d^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 522, normalized size of antiderivative = 2.58 \[ \int \frac {x^4 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (35 \, b^{2} c^{4} - 40 \, a b c^{3} d + 8 \, a^{2} c^{2} d^{2} + {\left (35 \, b^{2} c^{2} d^{2} - 40 \, a b c d^{3} + 8 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (35 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 8 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (6 \, b^{2} d^{4} x^{7} - 3 \, {\left (7 \, b^{2} c d^{3} - 8 \, a b d^{4}\right )} x^{5} - 4 \, {\left (35 \, b^{2} c^{2} d^{2} - 40 \, a b c d^{3} + 8 \, a^{2} d^{4}\right )} x^{3} - 3 \, {\left (35 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 8 \, a^{2} c d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{48 \, {\left (d^{7} x^{4} + 2 \, c d^{6} x^{2} + c^{2} d^{5}\right )}}, -\frac {3 \, {\left (35 \, b^{2} c^{4} - 40 \, a b c^{3} d + 8 \, a^{2} c^{2} d^{2} + {\left (35 \, b^{2} c^{2} d^{2} - 40 \, a b c d^{3} + 8 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (35 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 8 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - {\left (6 \, b^{2} d^{4} x^{7} - 3 \, {\left (7 \, b^{2} c d^{3} - 8 \, a b d^{4}\right )} x^{5} - 4 \, {\left (35 \, b^{2} c^{2} d^{2} - 40 \, a b c d^{3} + 8 \, a^{2} d^{4}\right )} x^{3} - 3 \, {\left (35 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 8 \, a^{2} c d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{24 \, {\left (d^{7} x^{4} + 2 \, c d^{6} x^{2} + c^{2} d^{5}\right )}}\right ] \]

[In]

integrate(x^4*(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(3*(35*b^2*c^4 - 40*a*b*c^3*d + 8*a^2*c^2*d^2 + (35*b^2*c^2*d^2 - 40*a*b*c*d^3 + 8*a^2*d^4)*x^4 + 2*(35*
b^2*c^3*d - 40*a*b*c^2*d^2 + 8*a^2*c*d^3)*x^2)*sqrt(d)*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) + 2*(6*
b^2*d^4*x^7 - 3*(7*b^2*c*d^3 - 8*a*b*d^4)*x^5 - 4*(35*b^2*c^2*d^2 - 40*a*b*c*d^3 + 8*a^2*d^4)*x^3 - 3*(35*b^2*
c^3*d - 40*a*b*c^2*d^2 + 8*a^2*c*d^3)*x)*sqrt(d*x^2 + c))/(d^7*x^4 + 2*c*d^6*x^2 + c^2*d^5), -1/24*(3*(35*b^2*
c^4 - 40*a*b*c^3*d + 8*a^2*c^2*d^2 + (35*b^2*c^2*d^2 - 40*a*b*c*d^3 + 8*a^2*d^4)*x^4 + 2*(35*b^2*c^3*d - 40*a*
b*c^2*d^2 + 8*a^2*c*d^3)*x^2)*sqrt(-d)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) - (6*b^2*d^4*x^7 - 3*(7*b^2*c*d^3 -
8*a*b*d^4)*x^5 - 4*(35*b^2*c^2*d^2 - 40*a*b*c*d^3 + 8*a^2*d^4)*x^3 - 3*(35*b^2*c^3*d - 40*a*b*c^2*d^2 + 8*a^2*
c*d^3)*x)*sqrt(d*x^2 + c))/(d^7*x^4 + 2*c*d^6*x^2 + c^2*d^5)]

Sympy [F]

\[ \int \frac {x^4 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {x^{4} \left (a + b x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(x**4*(b*x**2+a)**2/(d*x**2+c)**(5/2),x)

[Out]

Integral(x**4*(a + b*x**2)**2/(c + d*x**2)**(5/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.47 \[ \int \frac {x^4 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {b^{2} x^{7}}{4 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} d} - \frac {7 \, b^{2} c x^{5}}{8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} d^{2}} + \frac {a b x^{5}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d} - \frac {1}{3} \, a^{2} x {\left (\frac {3 \, x^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d} + \frac {2 \, c}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d^{2}}\right )} - \frac {35 \, b^{2} c^{2} x {\left (\frac {3 \, x^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d} + \frac {2 \, c}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d^{2}}\right )}}{24 \, d^{2}} + \frac {5 \, a b c x {\left (\frac {3 \, x^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d} + \frac {2 \, c}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d^{2}}\right )}}{3 \, d} - \frac {35 \, b^{2} c^{2} x}{24 \, \sqrt {d x^{2} + c} d^{4}} + \frac {5 \, a b c x}{3 \, \sqrt {d x^{2} + c} d^{3}} - \frac {a^{2} x}{3 \, \sqrt {d x^{2} + c} d^{2}} + \frac {35 \, b^{2} c^{2} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{8 \, d^{\frac {9}{2}}} - \frac {5 \, a b c \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{d^{\frac {7}{2}}} + \frac {a^{2} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{d^{\frac {5}{2}}} \]

[In]

integrate(x^4*(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

1/4*b^2*x^7/((d*x^2 + c)^(3/2)*d) - 7/8*b^2*c*x^5/((d*x^2 + c)^(3/2)*d^2) + a*b*x^5/((d*x^2 + c)^(3/2)*d) - 1/
3*a^2*x*(3*x^2/((d*x^2 + c)^(3/2)*d) + 2*c/((d*x^2 + c)^(3/2)*d^2)) - 35/24*b^2*c^2*x*(3*x^2/((d*x^2 + c)^(3/2
)*d) + 2*c/((d*x^2 + c)^(3/2)*d^2))/d^2 + 5/3*a*b*c*x*(3*x^2/((d*x^2 + c)^(3/2)*d) + 2*c/((d*x^2 + c)^(3/2)*d^
2))/d - 35/24*b^2*c^2*x/(sqrt(d*x^2 + c)*d^4) + 5/3*a*b*c*x/(sqrt(d*x^2 + c)*d^3) - 1/3*a^2*x/(sqrt(d*x^2 + c)
*d^2) + 35/8*b^2*c^2*arcsinh(d*x/sqrt(c*d))/d^(9/2) - 5*a*b*c*arcsinh(d*x/sqrt(c*d))/d^(7/2) + a^2*arcsinh(d*x
/sqrt(c*d))/d^(5/2)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.94 \[ \int \frac {x^4 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {{\left ({\left (3 \, {\left (\frac {2 \, b^{2} x^{2}}{d} - \frac {7 \, b^{2} c^{2} d^{5} - 8 \, a b c d^{6}}{c d^{7}}\right )} x^{2} - \frac {4 \, {\left (35 \, b^{2} c^{3} d^{4} - 40 \, a b c^{2} d^{5} + 8 \, a^{2} c d^{6}\right )}}{c d^{7}}\right )} x^{2} - \frac {3 \, {\left (35 \, b^{2} c^{4} d^{3} - 40 \, a b c^{3} d^{4} + 8 \, a^{2} c^{2} d^{5}\right )}}{c d^{7}}\right )} x}{24 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} - \frac {{\left (35 \, b^{2} c^{2} - 40 \, a b c d + 8 \, a^{2} d^{2}\right )} \log \left ({\left | -\sqrt {d} x + \sqrt {d x^{2} + c} \right |}\right )}{8 \, d^{\frac {9}{2}}} \]

[In]

integrate(x^4*(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="giac")

[Out]

1/24*((3*(2*b^2*x^2/d - (7*b^2*c^2*d^5 - 8*a*b*c*d^6)/(c*d^7))*x^2 - 4*(35*b^2*c^3*d^4 - 40*a*b*c^2*d^5 + 8*a^
2*c*d^6)/(c*d^7))*x^2 - 3*(35*b^2*c^4*d^3 - 40*a*b*c^3*d^4 + 8*a^2*c^2*d^5)/(c*d^7))*x/(d*x^2 + c)^(3/2) - 1/8
*(35*b^2*c^2 - 40*a*b*c*d + 8*a^2*d^2)*log(abs(-sqrt(d)*x + sqrt(d*x^2 + c)))/d^(9/2)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {x^4\,{\left (b\,x^2+a\right )}^2}{{\left (d\,x^2+c\right )}^{5/2}} \,d x \]

[In]

int((x^4*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x)

[Out]

int((x^4*(a + b*x^2)^2)/(c + d*x^2)^(5/2), x)