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

Optimal. Leaf size=223 \[ -\frac{a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}+\frac{x \left (c+d x^2\right )^{5/2} \left (4 a d (2 a d+3 b c)+b^2 c^2\right )}{6 c^2}+\frac{5 x \left (c+d x^2\right )^{3/2} \left (4 a d (2 a d+3 b c)+b^2 c^2\right )}{24 c}+\frac{5}{16} x \sqrt{c+d x^2} \left (4 a d (2 a d+3 b c)+b^2 c^2\right )+\frac{5 c \left (4 a d (2 a d+3 b c)+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 \sqrt{d}}-\frac{2 a \left (c+d x^2\right )^{7/2} (2 a d+3 b c)}{3 c^2 x} \]

[Out]

(5*(b^2*c^2 + 4*a*d*(3*b*c + 2*a*d))*x*Sqrt[c + d*x^2])/16 + (5*(b^2*c^2 + 4*a*d*(3*b*c + 2*a*d))*x*(c + d*x^2
)^(3/2))/(24*c) + ((b^2*c^2 + 4*a*d*(3*b*c + 2*a*d))*x*(c + d*x^2)^(5/2))/(6*c^2) - (a^2*(c + d*x^2)^(7/2))/(3
*c*x^3) - (2*a*(3*b*c + 2*a*d)*(c + d*x^2)^(7/2))/(3*c^2*x) + (5*c*(b^2*c^2 + 4*a*d*(3*b*c + 2*a*d))*ArcTanh[(
Sqrt[d]*x)/Sqrt[c + d*x^2]])/(16*Sqrt[d])

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Rubi [A]  time = 0.174919, antiderivative size = 219, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {462, 453, 195, 217, 206} \[ -\frac{a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}+\frac{1}{6} x \left (c+d x^2\right )^{5/2} \left (\frac{4 a d (2 a d+3 b c)}{c^2}+b^2\right )+\frac{5 x \left (c+d x^2\right )^{3/2} \left (4 a d (2 a d+3 b c)+b^2 c^2\right )}{24 c}+\frac{5}{16} x \sqrt{c+d x^2} \left (4 a d (2 a d+3 b c)+b^2 c^2\right )+\frac{5 c \left (4 a d (2 a d+3 b c)+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 \sqrt{d}}-\frac{2 a \left (c+d x^2\right )^{7/2} (2 a d+3 b c)}{3 c^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(b^2*c^2 + 4*a*d*(3*b*c + 2*a*d))*x*Sqrt[c + d*x^2])/16 + (5*(b^2*c^2 + 4*a*d*(3*b*c + 2*a*d))*x*(c + d*x^2
)^(3/2))/(24*c) + ((b^2 + (4*a*d*(3*b*c + 2*a*d))/c^2)*x*(c + d*x^2)^(5/2))/6 - (a^2*(c + d*x^2)^(7/2))/(3*c*x
^3) - (2*a*(3*b*c + 2*a*d)*(c + d*x^2)^(7/2))/(3*c^2*x) + (5*c*(b^2*c^2 + 4*a*d*(3*b*c + 2*a*d))*ArcTanh[(Sqrt
[d]*x)/Sqrt[c + d*x^2]])/(16*Sqrt[d])

Rule 462

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]

Rule 453

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

Rule 195

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 217

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 206

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

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^4} \, dx &=-\frac{a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}+\frac{\int \frac{\left (2 a (3 b c+2 a d)+3 b^2 c x^2\right ) \left (c+d x^2\right )^{5/2}}{x^2} \, dx}{3 c}\\ &=-\frac{a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}-\frac{2 a (3 b c+2 a d) \left (c+d x^2\right )^{7/2}}{3 c^2 x}+\left (b^2+\frac{4 a d (3 b c+2 a d)}{c^2}\right ) \int \left (c+d x^2\right )^{5/2} \, dx\\ &=\frac{1}{6} \left (b^2+\frac{4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{5/2}-\frac{a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}-\frac{2 a (3 b c+2 a d) \left (c+d x^2\right )^{7/2}}{3 c^2 x}+\frac{1}{6} \left (5 c \left (b^2+\frac{4 a d (3 b c+2 a d)}{c^2}\right )\right ) \int \left (c+d x^2\right )^{3/2} \, dx\\ &=\frac{5}{24} c \left (b^2+\frac{4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{3/2}+\frac{1}{6} \left (b^2+\frac{4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{5/2}-\frac{a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}-\frac{2 a (3 b c+2 a d) \left (c+d x^2\right )^{7/2}}{3 c^2 x}+\frac{1}{8} \left (5 \left (b^2 c^2+12 a b c d+8 a^2 d^2\right )\right ) \int \sqrt{c+d x^2} \, dx\\ &=\frac{5}{16} \left (b^2 c^2+12 a b c d+8 a^2 d^2\right ) x \sqrt{c+d x^2}+\frac{5}{24} c \left (b^2+\frac{4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{3/2}+\frac{1}{6} \left (b^2+\frac{4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{5/2}-\frac{a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}-\frac{2 a (3 b c+2 a d) \left (c+d x^2\right )^{7/2}}{3 c^2 x}+\frac{1}{16} \left (5 c \left (b^2 c^2+12 a b c d+8 a^2 d^2\right )\right ) \int \frac{1}{\sqrt{c+d x^2}} \, dx\\ &=\frac{5}{16} \left (b^2 c^2+12 a b c d+8 a^2 d^2\right ) x \sqrt{c+d x^2}+\frac{5}{24} c \left (b^2+\frac{4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{3/2}+\frac{1}{6} \left (b^2+\frac{4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{5/2}-\frac{a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}-\frac{2 a (3 b c+2 a d) \left (c+d x^2\right )^{7/2}}{3 c^2 x}+\frac{1}{16} \left (5 c \left (b^2 c^2+12 a b c d+8 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-d x^2} \, dx,x,\frac{x}{\sqrt{c+d x^2}}\right )\\ &=\frac{5}{16} \left (b^2 c^2+12 a b c d+8 a^2 d^2\right ) x \sqrt{c+d x^2}+\frac{5}{24} c \left (b^2+\frac{4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{3/2}+\frac{1}{6} \left (b^2+\frac{4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{5/2}-\frac{a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}-\frac{2 a (3 b c+2 a d) \left (c+d x^2\right )^{7/2}}{3 c^2 x}+\frac{5 c \left (b^2 c^2+12 a b c d+8 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 \sqrt{d}}\\ \end{align*}

Mathematica [A]  time = 0.119293, size = 155, normalized size = 0.7 \[ \frac{1}{48} \left (\frac{\sqrt{c+d x^2} \left (-8 a^2 \left (2 c^2+14 c d x^2-3 d^2 x^4\right )+12 a b x^2 \left (-8 c^2+9 c d x^2+2 d^2 x^4\right )+b^2 x^4 \left (33 c^2+26 c d x^2+8 d^2 x^4\right )\right )}{x^3}+\frac{15 c \left (8 a^2 d^2+12 a b c d+b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{\sqrt{d}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[c + d*x^2]*(-8*a^2*(2*c^2 + 14*c*d*x^2 - 3*d^2*x^4) + 12*a*b*x^2*(-8*c^2 + 9*c*d*x^2 + 2*d^2*x^4) + b^2
*x^4*(33*c^2 + 26*c*d*x^2 + 8*d^2*x^4)))/x^3 + (15*c*(b^2*c^2 + 12*a*b*c*d + 8*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt
[c + d*x^2]])/Sqrt[d])/48

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Maple [A]  time = 0.012, size = 298, normalized size = 1.3 \begin{align*}{\frac{x{b}^{2}}{6} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{b}^{2}cx}{24} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{2}{c}^{2}x}{16}\sqrt{d{x}^{2}+c}}+{\frac{5\,{b}^{2}{c}^{3}}{16}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){\frac{1}{\sqrt{d}}}}-{\frac{{a}^{2}}{3\,c{x}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{4\,{a}^{2}d}{3\,{c}^{2}x} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{4\,{a}^{2}{d}^{2}x}{3\,{c}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{a}^{2}{d}^{2}x}{3\,c} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}{d}^{2}x}{2}\sqrt{d{x}^{2}+c}}+{\frac{5\,{a}^{2}c}{2}{d}^{{\frac{3}{2}}}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) }-2\,{\frac{ab \left ( d{x}^{2}+c \right ) ^{7/2}}{cx}}+2\,{\frac{abdx \left ( d{x}^{2}+c \right ) ^{5/2}}{c}}+{\frac{5\,abdx}{2} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{15\,cabdx}{4}\sqrt{d{x}^{2}+c}}+{\frac{15\,ab{c}^{2}}{4}\sqrt{d}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x*b^2*(d*x^2+c)^(5/2)+5/24*b^2*c*x*(d*x^2+c)^(3/2)+5/16*b^2*c^2*x*(d*x^2+c)^(1/2)+5/16*b^2*c^3/d^(1/2)*ln(
x*d^(1/2)+(d*x^2+c)^(1/2))-1/3*a^2*(d*x^2+c)^(7/2)/c/x^3-4/3*a^2*d/c^2/x*(d*x^2+c)^(7/2)+4/3*a^2*d^2/c^2*x*(d*
x^2+c)^(5/2)+5/3*a^2*d^2/c*x*(d*x^2+c)^(3/2)+5/2*a^2*d^2*x*(d*x^2+c)^(1/2)+5/2*a^2*d^(3/2)*c*ln(x*d^(1/2)+(d*x
^2+c)^(1/2))-2*a*b/c/x*(d*x^2+c)^(7/2)+2*a*b*d/c*x*(d*x^2+c)^(5/2)+5/2*a*b*d*x*(d*x^2+c)^(3/2)+15/4*a*b*d*c*x*
(d*x^2+c)^(1/2)+15/4*a*b*d^(1/2)*c^2*ln(x*d^(1/2)+(d*x^2+c)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.52664, size = 776, normalized size = 3.48 \begin{align*} \left [\frac{15 \,{\left (b^{2} c^{3} + 12 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} \sqrt{d} x^{3} \log \left (-2 \, d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{d} x - c\right ) + 2 \,{\left (8 \, b^{2} d^{3} x^{8} + 2 \,{\left (13 \, b^{2} c d^{2} + 12 \, a b d^{3}\right )} x^{6} - 16 \, a^{2} c^{2} d + 3 \,{\left (11 \, b^{2} c^{2} d + 36 \, a b c d^{2} + 8 \, a^{2} d^{3}\right )} x^{4} - 16 \,{\left (6 \, a b c^{2} d + 7 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{96 \, d x^{3}}, -\frac{15 \,{\left (b^{2} c^{3} + 12 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} \sqrt{-d} x^{3} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right ) -{\left (8 \, b^{2} d^{3} x^{8} + 2 \,{\left (13 \, b^{2} c d^{2} + 12 \, a b d^{3}\right )} x^{6} - 16 \, a^{2} c^{2} d + 3 \,{\left (11 \, b^{2} c^{2} d + 36 \, a b c d^{2} + 8 \, a^{2} d^{3}\right )} x^{4} - 16 \,{\left (6 \, a b c^{2} d + 7 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{48 \, d x^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/96*(15*(b^2*c^3 + 12*a*b*c^2*d + 8*a^2*c*d^2)*sqrt(d)*x^3*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) +
2*(8*b^2*d^3*x^8 + 2*(13*b^2*c*d^2 + 12*a*b*d^3)*x^6 - 16*a^2*c^2*d + 3*(11*b^2*c^2*d + 36*a*b*c*d^2 + 8*a^2*
d^3)*x^4 - 16*(6*a*b*c^2*d + 7*a^2*c*d^2)*x^2)*sqrt(d*x^2 + c))/(d*x^3), -1/48*(15*(b^2*c^3 + 12*a*b*c^2*d + 8
*a^2*c*d^2)*sqrt(-d)*x^3*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) - (8*b^2*d^3*x^8 + 2*(13*b^2*c*d^2 + 12*a*b*d^3)*x
^6 - 16*a^2*c^2*d + 3*(11*b^2*c^2*d + 36*a*b*c*d^2 + 8*a^2*d^3)*x^4 - 16*(6*a*b*c^2*d + 7*a^2*c*d^2)*x^2)*sqrt
(d*x^2 + c))/(d*x^3)]

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Sympy [B]  time = 20.9042, size = 490, normalized size = 2.2 \begin{align*} - \frac{2 a^{2} c^{\frac{3}{2}} d}{x \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{a^{2} \sqrt{c} d^{2} x \sqrt{1 + \frac{d x^{2}}{c}}}{2} - \frac{2 a^{2} \sqrt{c} d^{2} x}{\sqrt{1 + \frac{d x^{2}}{c}}} - \frac{a^{2} c^{2} \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{3 x^{2}} - \frac{a^{2} c d^{\frac{3}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{3} + \frac{5 a^{2} c d^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{2} - \frac{2 a b c^{\frac{5}{2}}}{x \sqrt{1 + \frac{d x^{2}}{c}}} + 2 a b c^{\frac{3}{2}} d x \sqrt{1 + \frac{d x^{2}}{c}} - \frac{7 a b c^{\frac{3}{2}} d x}{4 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 a b \sqrt{c} d^{2} x^{3}}{4 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{15 a b c^{2} \sqrt{d} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{4} + \frac{a b d^{3} x^{5}}{2 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{b^{2} c^{\frac{5}{2}} x \sqrt{1 + \frac{d x^{2}}{c}}}{2} + \frac{3 b^{2} c^{\frac{5}{2}} x}{16 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{35 b^{2} c^{\frac{3}{2}} d x^{3}}{48 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{17 b^{2} \sqrt{c} d^{2} x^{5}}{24 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 b^{2} c^{3} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{16 \sqrt{d}} + \frac{b^{2} d^{3} x^{7}}{6 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**2*c**(3/2)*d/(x*sqrt(1 + d*x**2/c)) + a**2*sqrt(c)*d**2*x*sqrt(1 + d*x**2/c)/2 - 2*a**2*sqrt(c)*d**2*x/s
qrt(1 + d*x**2/c) - a**2*c**2*sqrt(d)*sqrt(c/(d*x**2) + 1)/(3*x**2) - a**2*c*d**(3/2)*sqrt(c/(d*x**2) + 1)/3 +
5*a**2*c*d**(3/2)*asinh(sqrt(d)*x/sqrt(c))/2 - 2*a*b*c**(5/2)/(x*sqrt(1 + d*x**2/c)) + 2*a*b*c**(3/2)*d*x*sqr
t(1 + d*x**2/c) - 7*a*b*c**(3/2)*d*x/(4*sqrt(1 + d*x**2/c)) + 3*a*b*sqrt(c)*d**2*x**3/(4*sqrt(1 + d*x**2/c)) +
15*a*b*c**2*sqrt(d)*asinh(sqrt(d)*x/sqrt(c))/4 + a*b*d**3*x**5/(2*sqrt(c)*sqrt(1 + d*x**2/c)) + b**2*c**(5/2)
*x*sqrt(1 + d*x**2/c)/2 + 3*b**2*c**(5/2)*x/(16*sqrt(1 + d*x**2/c)) + 35*b**2*c**(3/2)*d*x**3/(48*sqrt(1 + d*x
**2/c)) + 17*b**2*sqrt(c)*d**2*x**5/(24*sqrt(1 + d*x**2/c)) + 5*b**2*c**3*asinh(sqrt(d)*x/sqrt(c))/(16*sqrt(d)
) + b**2*d**3*x**7/(6*sqrt(c)*sqrt(1 + d*x**2/c))

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Giac [A]  time = 1.18788, size = 414, normalized size = 1.86 \begin{align*} \frac{1}{48} \,{\left (2 \,{\left (4 \, b^{2} d^{2} x^{2} + \frac{13 \, b^{2} c d^{5} + 12 \, a b d^{6}}{d^{4}}\right )} x^{2} + \frac{3 \,{\left (11 \, b^{2} c^{2} d^{4} + 36 \, a b c d^{5} + 8 \, a^{2} d^{6}\right )}}{d^{4}}\right )} \sqrt{d x^{2} + c} x - \frac{5 \,{\left (b^{2} c^{3} \sqrt{d} + 12 \, a b c^{2} d^{\frac{3}{2}} + 8 \, a^{2} c d^{\frac{5}{2}}\right )} \log \left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{32 \, d} + \frac{2 \,{\left (6 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a b c^{3} \sqrt{d} + 9 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a^{2} c^{2} d^{\frac{3}{2}} - 12 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b c^{4} \sqrt{d} - 12 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a^{2} c^{3} d^{\frac{3}{2}} + 6 \, a b c^{5} \sqrt{d} + 7 \, a^{2} c^{4} d^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(2*(4*b^2*d^2*x^2 + (13*b^2*c*d^5 + 12*a*b*d^6)/d^4)*x^2 + 3*(11*b^2*c^2*d^4 + 36*a*b*c*d^5 + 8*a^2*d^6)/
d^4)*sqrt(d*x^2 + c)*x - 5/32*(b^2*c^3*sqrt(d) + 12*a*b*c^2*d^(3/2) + 8*a^2*c*d^(5/2))*log((sqrt(d)*x - sqrt(d
*x^2 + c))^2)/d + 2/3*(6*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a*b*c^3*sqrt(d) + 9*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a
^2*c^2*d^(3/2) - 12*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*b*c^4*sqrt(d) - 12*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^2*c
^3*d^(3/2) + 6*a*b*c^5*sqrt(d) + 7*a^2*c^4*d^(3/2))/((sqrt(d)*x - sqrt(d*x^2 + c))^2 - c)^3