\(\int \frac {(a+b x^2)^2 (c+d x^2)^{5/2}}{x^2} \, dx\) [630]

Optimal result

Integrand size = 24, antiderivative size = 217 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^2} \, dx=-\frac {5 c \left (b^2 c^2-16 a d (b c+3 a d)\right ) x \sqrt {c+d x^2}}{128 d}-\frac {5 \left (b^2 c^2-16 a d (b c+3 a d)\right ) x \left (c+d x^2\right )^{3/2}}{192 d}-\frac {\left (b^2 c^2-16 a d (b c+3 a d)\right ) x \left (c+d x^2\right )^{5/2}}{48 c d}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{c x}+\frac {b^2 x \left (c+d x^2\right )^{7/2}}{8 d}-\frac {5 c^2 \left (b^2 c^2-16 a d (b c+3 a d)\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{128 d^{3/2}} \]

[Out]

-5/192*(b^2*c^2-16*a*d*(3*a*d+b*c))*x*(d*x^2+c)^(3/2)/d-1/48*(b^2*c^2-16*a*d*(3*a*d+b*c))*x*(d*x^2+c)^(5/2)/c/
d-a^2*(d*x^2+c)^(7/2)/c/x+1/8*b^2*x*(d*x^2+c)^(7/2)/d-5/128*c^2*(b^2*c^2-16*a*d*(3*a*d+b*c))*arctanh(x*d^(1/2)
/(d*x^2+c)^(1/2))/d^(3/2)-5/128*c*(b^2*c^2-16*a*d*(3*a*d+b*c))*x*(d*x^2+c)^(1/2)/d

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {473, 396, 201, 223, 212} \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^2} \, dx=-\frac {a^2 \left (c+d x^2\right )^{7/2}}{c x}-\frac {5 c^2 \left (b^2 c^2-16 a d (3 a d+b c)\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{128 d^{3/2}}-\frac {5 x \left (c+d x^2\right )^{3/2} \left (b^2 c^2-16 a d (3 a d+b c)\right )}{192 d}-\frac {5 c x \sqrt {c+d x^2} \left (b^2 c^2-16 a d (3 a d+b c)\right )}{128 d}-\frac {1}{48} x \left (c+d x^2\right )^{5/2} \left (\frac {b^2 c}{d}-\frac {16 a (3 a d+b c)}{c}\right )+\frac {b^2 x \left (c+d x^2\right )^{7/2}}{8 d} \]

[In]

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

[Out]

(-5*c*(b^2*c^2 - 16*a*d*(b*c + 3*a*d))*x*Sqrt[c + d*x^2])/(128*d) - (5*(b^2*c^2 - 16*a*d*(b*c + 3*a*d))*x*(c +
d*x^2)^(3/2))/(192*d) - (((b^2*c)/d - (16*a*(b*c + 3*a*d))/c)*x*(c + d*x^2)^(5/2))/48 - (a^2*(c + d*x^2)^(7/2
))/(c*x) + (b^2*x*(c + d*x^2)^(7/2))/(8*d) - (5*c^2*(b^2*c^2 - 16*a*d*(b*c + 3*a*d))*ArcTanh[(Sqrt[d]*x)/Sqrt[
c + d*x^2]])/(128*d^(3/2))

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 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 473

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

Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 \left (c+d x^2\right )^{7/2}}{c x}+\frac {\int \left (2 a (b c+3 a d)+b^2 c x^2\right ) \left (c+d x^2\right )^{5/2} \, dx}{c} \\ & = -\frac {a^2 \left (c+d x^2\right )^{7/2}}{c x}+\frac {b^2 x \left (c+d x^2\right )^{7/2}}{8 d}-\frac {\left (b^2 c^2-16 a d (b c+3 a d)\right ) \int \left (c+d x^2\right )^{5/2} \, dx}{8 c d} \\ & = -\frac {1}{48} \left (\frac {b^2 c}{d}-\frac {16 a (b c+3 a d)}{c}\right ) x \left (c+d x^2\right )^{5/2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{c x}+\frac {b^2 x \left (c+d x^2\right )^{7/2}}{8 d}-\frac {\left (5 \left (b^2 c^2-16 a d (b c+3 a d)\right )\right ) \int \left (c+d x^2\right )^{3/2} \, dx}{48 d} \\ & = -\frac {5 \left (b^2 c^2-16 a d (b c+3 a d)\right ) x \left (c+d x^2\right )^{3/2}}{192 d}-\frac {1}{48} \left (\frac {b^2 c}{d}-\frac {16 a (b c+3 a d)}{c}\right ) x \left (c+d x^2\right )^{5/2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{c x}+\frac {b^2 x \left (c+d x^2\right )^{7/2}}{8 d}-\frac {\left (5 c \left (b^2 c^2-16 a d (b c+3 a d)\right )\right ) \int \sqrt {c+d x^2} \, dx}{64 d} \\ & = -\frac {5 c \left (b^2 c^2-16 a d (b c+3 a d)\right ) x \sqrt {c+d x^2}}{128 d}-\frac {5 \left (b^2 c^2-16 a d (b c+3 a d)\right ) x \left (c+d x^2\right )^{3/2}}{192 d}-\frac {1}{48} \left (\frac {b^2 c}{d}-\frac {16 a (b c+3 a d)}{c}\right ) x \left (c+d x^2\right )^{5/2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{c x}+\frac {b^2 x \left (c+d x^2\right )^{7/2}}{8 d}-\frac {\left (5 c^2 \left (b^2 c^2-16 a d (b c+3 a d)\right )\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{128 d} \\ & = -\frac {5 c \left (b^2 c^2-16 a d (b c+3 a d)\right ) x \sqrt {c+d x^2}}{128 d}-\frac {5 \left (b^2 c^2-16 a d (b c+3 a d)\right ) x \left (c+d x^2\right )^{3/2}}{192 d}-\frac {1}{48} \left (\frac {b^2 c}{d}-\frac {16 a (b c+3 a d)}{c}\right ) x \left (c+d x^2\right )^{5/2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{c x}+\frac {b^2 x \left (c+d x^2\right )^{7/2}}{8 d}-\frac {\left (5 c^2 \left (b^2 c^2-16 a d (b c+3 a d)\right )\right ) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{128 d} \\ & = -\frac {5 c \left (b^2 c^2-16 a d (b c+3 a d)\right ) x \sqrt {c+d x^2}}{128 d}-\frac {5 \left (b^2 c^2-16 a d (b c+3 a d)\right ) x \left (c+d x^2\right )^{3/2}}{192 d}-\frac {1}{48} \left (\frac {b^2 c}{d}-\frac {16 a (b c+3 a d)}{c}\right ) x \left (c+d x^2\right )^{5/2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{c x}+\frac {b^2 x \left (c+d x^2\right )^{7/2}}{8 d}-\frac {5 c^2 \left (b^2 c^2-16 a d (b c+3 a d)\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{128 d^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^2} \, dx=\frac {\sqrt {d} \sqrt {c+d x^2} \left (48 a^2 d \left (-8 c^2+9 c d x^2+2 d^2 x^4\right )+16 a b d x^2 \left (33 c^2+26 c d x^2+8 d^2 x^4\right )+b^2 x^2 \left (15 c^3+118 c^2 d x^2+136 c d^2 x^4+48 d^3 x^6\right )\right )+15 c^2 \left (b^2 c^2-16 a b c d-48 a^2 d^2\right ) x \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )}{384 d^{3/2} x} \]

[In]

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

[Out]

(Sqrt[d]*Sqrt[c + d*x^2]*(48*a^2*d*(-8*c^2 + 9*c*d*x^2 + 2*d^2*x^4) + 16*a*b*d*x^2*(33*c^2 + 26*c*d*x^2 + 8*d^
2*x^4) + b^2*x^2*(15*c^3 + 118*c^2*d*x^2 + 136*c*d^2*x^4 + 48*d^3*x^6)) + 15*c^2*(b^2*c^2 - 16*a*b*c*d - 48*a^
2*d^2)*x*Log[-(Sqrt[d]*x) + Sqrt[c + d*x^2]])/(384*d^(3/2)*x)

Maple [A] (verified)

Time = 2.98 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.74

method	result	size
pseudoelliptic	\(\frac {\frac {15 x \,c^{2} \left (a^{2} d^{2}+\frac {1}{3} a b c d -\frac {1}{48} b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right )}{8}+\frac {9 \sqrt {d \,x^{2}+c}\, \left (-\frac {8 \left (-\frac {59}{192} b^{2} x^{4}-\frac {11}{8} a b \,x^{2}+a^{2}\right ) c^{2} d^{\frac {3}{2}}}{9}+x^{2} \left (c \left (\frac {17}{54} b^{2} x^{4}+\frac {26}{27} a b \,x^{2}+a^{2}\right ) d^{\frac {5}{2}}+\frac {\left (b^{2} x^{6}+\frac {8}{3} a b \,x^{4}+2 a^{2} x^{2}\right ) d^{\frac {7}{2}}}{9}+\frac {5 b^{2} c^{3} \sqrt {d}}{144}\right )\right )}{8}}{x \,d^{\frac {3}{2}}}\)	\(161\)
risch	\(-\frac {\sqrt {d \,x^{2}+c}\, \left (-48 b^{2} d^{3} x^{8}-128 a b \,d^{3} x^{6}-136 b^{2} c \,d^{2} x^{6}-96 a^{2} d^{3} x^{4}-416 a b c \,d^{2} x^{4}-118 b^{2} c^{2} d \,x^{4}-432 a^{2} c \,d^{2} x^{2}-528 a b \,c^{2} d \,x^{2}-15 b^{2} c^{3} x^{2}+384 a^{2} c^{2} d \right )}{384 d x}+\frac {5 c^{2} \left (48 a^{2} d^{2}+16 a b c d -b^{2} c^{2}\right ) \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{128 d^{\frac {3}{2}}}\)	\(177\)
default	\(b^{2} \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{8 d}-\frac {c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6}\right )}{8 d}\right )+a^{2} \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}}}{c x}+\frac {6 d \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6}\right )}{c}\right )+2 a b \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6}\right )\)	\(261\)

[In]

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

[Out]

9/8/d^(3/2)*(5/3*x*c^2*(a^2*d^2+1/3*a*b*c*d-1/48*b^2*c^2)*arctanh((d*x^2+c)^(1/2)/x/d^(1/2))+(d*x^2+c)^(1/2)*(
-8/9*(-59/192*b^2*x^4-11/8*a*b*x^2+a^2)*c^2*d^(3/2)+x^2*(c*(17/54*b^2*x^4+26/27*a*b*x^2+a^2)*d^(5/2)+1/9*(b^2*
x^6+8/3*a*b*x^4+2*a^2*x^2)*d^(7/2)+5/144*b^2*c^3*d^(1/2))))/x

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.73 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^2} \, dx=\left [-\frac {15 \, {\left (b^{2} c^{4} - 16 \, a b c^{3} d - 48 \, a^{2} c^{2} d^{2}\right )} \sqrt {d} x \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - 2 \, {\left (48 \, b^{2} d^{4} x^{8} + 8 \, {\left (17 \, b^{2} c d^{3} + 16 \, a b d^{4}\right )} x^{6} - 384 \, a^{2} c^{2} d^{2} + 2 \, {\left (59 \, b^{2} c^{2} d^{2} + 208 \, a b c d^{3} + 48 \, a^{2} d^{4}\right )} x^{4} + 3 \, {\left (5 \, b^{2} c^{3} d + 176 \, a b c^{2} d^{2} + 144 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{768 \, d^{2} x}, \frac {15 \, {\left (b^{2} c^{4} - 16 \, a b c^{3} d - 48 \, a^{2} c^{2} d^{2}\right )} \sqrt {-d} x \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (48 \, b^{2} d^{4} x^{8} + 8 \, {\left (17 \, b^{2} c d^{3} + 16 \, a b d^{4}\right )} x^{6} - 384 \, a^{2} c^{2} d^{2} + 2 \, {\left (59 \, b^{2} c^{2} d^{2} + 208 \, a b c d^{3} + 48 \, a^{2} d^{4}\right )} x^{4} + 3 \, {\left (5 \, b^{2} c^{3} d + 176 \, a b c^{2} d^{2} + 144 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{384 \, d^{2} x}\right ] \]

[In]

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

[Out]

[-1/768*(15*(b^2*c^4 - 16*a*b*c^3*d - 48*a^2*c^2*d^2)*sqrt(d)*x*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c
) - 2*(48*b^2*d^4*x^8 + 8*(17*b^2*c*d^3 + 16*a*b*d^4)*x^6 - 384*a^2*c^2*d^2 + 2*(59*b^2*c^2*d^2 + 208*a*b*c*d^
3 + 48*a^2*d^4)*x^4 + 3*(5*b^2*c^3*d + 176*a*b*c^2*d^2 + 144*a^2*c*d^3)*x^2)*sqrt(d*x^2 + c))/(d^2*x), 1/384*(
15*(b^2*c^4 - 16*a*b*c^3*d - 48*a^2*c^2*d^2)*sqrt(-d)*x*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) + (48*b^2*d^4*x^8 +
8*(17*b^2*c*d^3 + 16*a*b*d^4)*x^6 - 384*a^2*c^2*d^2 + 2*(59*b^2*c^2*d^2 + 208*a*b*c*d^3 + 48*a^2*d^4)*x^4 + 3
*(5*b^2*c^3*d + 176*a*b*c^2*d^2 + 144*a^2*c*d^3)*x^2)*sqrt(d*x^2 + c))/(d^2*x)]

Sympy [A] (verification not implemented)

Time = 2.58 (sec) , antiderivative size = 915, normalized size of antiderivative = 4.22 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-a**2*c**(5/2)/(x*sqrt(1 + d*x**2/c)) - a**2*c**(3/2)*d*x/sqrt(1 + d*x**2/c) + a**2*c**2*sqrt(d)*asinh(sqrt(d)
*x/sqrt(c)) + 2*a**2*c*d*Piecewise((c*Piecewise((log(2*sqrt(d)*sqrt(c + d*x**2) + 2*d*x)/sqrt(d), Ne(c, 0)), (
x*log(x)/sqrt(d*x**2), True))/2 + x*sqrt(c + d*x**2)/2, Ne(d, 0)), (sqrt(c)*x, True)) + a**2*d**2*Piecewise((-
c**2*Piecewise((log(2*sqrt(d)*sqrt(c + d*x**2) + 2*d*x)/sqrt(d), Ne(c, 0)), (x*log(x)/sqrt(d*x**2), True))/(8*
d) + c*x*sqrt(c + d*x**2)/(8*d) + x**3*sqrt(c + d*x**2)/4, Ne(d, 0)), (sqrt(c)*x**3/3, True)) + 2*a*b*c**2*Pie
cewise((c*Piecewise((log(2*sqrt(d)*sqrt(c + d*x**2) + 2*d*x)/sqrt(d), Ne(c, 0)), (x*log(x)/sqrt(d*x**2), True)
)/2 + x*sqrt(c + d*x**2)/2, Ne(d, 0)), (sqrt(c)*x, True)) + 4*a*b*c*d*Piecewise((-c**2*Piecewise((log(2*sqrt(d
)*sqrt(c + d*x**2) + 2*d*x)/sqrt(d), Ne(c, 0)), (x*log(x)/sqrt(d*x**2), True))/(8*d) + c*x*sqrt(c + d*x**2)/(8
*d) + x**3*sqrt(c + d*x**2)/4, Ne(d, 0)), (sqrt(c)*x**3/3, True)) + 2*a*b*d**2*Piecewise((c**3*Piecewise((log(
2*sqrt(d)*sqrt(c + d*x**2) + 2*d*x)/sqrt(d), Ne(c, 0)), (x*log(x)/sqrt(d*x**2), True))/(16*d**2) - c**2*x*sqrt
(c + d*x**2)/(16*d**2) + c*x**3*sqrt(c + d*x**2)/(24*d) + x**5*sqrt(c + d*x**2)/6, Ne(d, 0)), (sqrt(c)*x**5/5,
True)) + b**2*c**2*Piecewise((-c**2*Piecewise((log(2*sqrt(d)*sqrt(c + d*x**2) + 2*d*x)/sqrt(d), Ne(c, 0)), (x
*log(x)/sqrt(d*x**2), True))/(8*d) + c*x*sqrt(c + d*x**2)/(8*d) + x**3*sqrt(c + d*x**2)/4, Ne(d, 0)), (sqrt(c)
*x**3/3, True)) + 2*b**2*c*d*Piecewise((c**3*Piecewise((log(2*sqrt(d)*sqrt(c + d*x**2) + 2*d*x)/sqrt(d), Ne(c,
0)), (x*log(x)/sqrt(d*x**2), True))/(16*d**2) - c**2*x*sqrt(c + d*x**2)/(16*d**2) + c*x**3*sqrt(c + d*x**2)/(
24*d) + x**5*sqrt(c + d*x**2)/6, Ne(d, 0)), (sqrt(c)*x**5/5, True)) + b**2*d**2*Piecewise((-5*c**4*Piecewise((
log(2*sqrt(d)*sqrt(c + d*x**2) + 2*d*x)/sqrt(d), Ne(c, 0)), (x*log(x)/sqrt(d*x**2), True))/(128*d**3) + 5*c**3
*x*sqrt(c + d*x**2)/(128*d**3) - 5*c**2*x**3*sqrt(c + d*x**2)/(192*d**2) + c*x**5*sqrt(c + d*x**2)/(48*d) + x*
*7*sqrt(c + d*x**2)/8, Ne(d, 0)), (sqrt(c)*x**7/7, True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^2} \, dx=\frac {1}{3} \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b x + \frac {5}{12} \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c x + \frac {5}{8} \, \sqrt {d x^{2} + c} a b c^{2} x + \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} x}{8 \, d} - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c x}{48 \, d} - \frac {5 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{2} x}{192 \, d} - \frac {5 \, \sqrt {d x^{2} + c} b^{2} c^{3} x}{128 \, d} + \frac {5}{4} \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d x + \frac {15}{8} \, \sqrt {d x^{2} + c} a^{2} c d x - \frac {5 \, b^{2} c^{4} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{128 \, d^{\frac {3}{2}}} + \frac {5 \, a b c^{3} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{8 \, \sqrt {d}} + \frac {15}{8} \, a^{2} c^{2} \sqrt {d} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right ) - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2}}{x} \]

[In]

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

[Out]

1/3*(d*x^2 + c)^(5/2)*a*b*x + 5/12*(d*x^2 + c)^(3/2)*a*b*c*x + 5/8*sqrt(d*x^2 + c)*a*b*c^2*x + 1/8*(d*x^2 + c)
^(7/2)*b^2*x/d - 1/48*(d*x^2 + c)^(5/2)*b^2*c*x/d - 5/192*(d*x^2 + c)^(3/2)*b^2*c^2*x/d - 5/128*sqrt(d*x^2 + c
)*b^2*c^3*x/d + 5/4*(d*x^2 + c)^(3/2)*a^2*d*x + 15/8*sqrt(d*x^2 + c)*a^2*c*d*x - 5/128*b^2*c^4*arcsinh(d*x/sqr
t(c*d))/d^(3/2) + 5/8*a*b*c^3*arcsinh(d*x/sqrt(c*d))/sqrt(d) + 15/8*a^2*c^2*sqrt(d)*arcsinh(d*x/sqrt(c*d)) - (
d*x^2 + c)^(5/2)*a^2/x

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^2} \, dx=\frac {2 \, a^{2} c^{3} \sqrt {d}}{{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c} + \frac {1}{384} \, {\left (2 \, {\left (4 \, {\left (6 \, b^{2} d^{2} x^{2} + \frac {17 \, b^{2} c d^{7} + 16 \, a b d^{8}}{d^{6}}\right )} x^{2} + \frac {59 \, b^{2} c^{2} d^{6} + 208 \, a b c d^{7} + 48 \, a^{2} d^{8}}{d^{6}}\right )} x^{2} + \frac {3 \, {\left (5 \, b^{2} c^{3} d^{5} + 176 \, a b c^{2} d^{6} + 144 \, a^{2} c d^{7}\right )}}{d^{6}}\right )} \sqrt {d x^{2} + c} x + \frac {5 \, {\left (b^{2} c^{4} - 16 \, a b c^{3} d - 48 \, a^{2} c^{2} d^{2}\right )} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{256 \, d^{\frac {3}{2}}} \]

[In]

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

[Out]

2*a^2*c^3*sqrt(d)/((sqrt(d)*x - sqrt(d*x^2 + c))^2 - c) + 1/384*(2*(4*(6*b^2*d^2*x^2 + (17*b^2*c*d^7 + 16*a*b*
d^8)/d^6)*x^2 + (59*b^2*c^2*d^6 + 208*a*b*c*d^7 + 48*a^2*d^8)/d^6)*x^2 + 3*(5*b^2*c^3*d^5 + 176*a*b*c^2*d^6 +
144*a^2*c*d^7)/d^6)*sqrt(d*x^2 + c)*x + 5/256*(b^2*c^4 - 16*a*b*c^3*d - 48*a^2*c^2*d^2)*log((sqrt(d)*x - sqrt(
d*x^2 + c))^2)/d^(3/2)

Mupad [F(-1)]

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

[In]

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

[Out]

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