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

Optimal. Leaf size=217 \[ -\frac{a^2 \left (c+d x^2\right )^{7/2}}{c x}-\frac{5 c^2 \left (b^2 c^2-16 a d (3 a d+b c)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{128 d^{3/2}}-\frac{x \left (c+d x^2\right )^{5/2} \left (b^2 c^2-16 a d (3 a d+b c)\right )}{48 c d}-\frac{5 x \left (c+d x^2\right )^{3/2} \left (b^2 c^2-16 a d (3 a d+b c)\right )}{192 d}-\frac{5 c x \sqrt{c+d x^2} \left (b^2 c^2-16 a d (3 a d+b c)\right )}{128 d}+\frac{b^2 x \left (c+d x^2\right )^{7/2}}{8 d} \]

[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^2 - 16*a*d*(b*c + 3*a*d))*x*(c + d*x^2)^(5/2))/(48*c*d) - (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)/Sqr
t[c + d*x^2]])/(128*d^(3/2))

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Rubi [A]  time = 0.141456, antiderivative size = 214, normalized size of antiderivative = 0.99, 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, 388, 195, 217, 206} \[ -\frac{a^2 \left (c+d x^2\right )^{7/2}}{c x}-\frac{5 c^2 \left (b^2 c^2-16 a d (3 a d+b c)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{128 d^{3/2}}-\frac{5 x \left (c+d x^2\right )^{3/2} \left (b^2 c^2-16 a d (3 a d+b c)\right )}{192 d}-\frac{5 c x \sqrt{c+d x^2} \left (b^2 c^2-16 a d (3 a d+b c)\right )}{128 d}-\frac{1}{48} x \left (c+d x^2\right )^{5/2} \left (\frac{b^2 c}{d}-\frac{16 a (3 a d+b c)}{c}\right )+\frac{b^2 x \left (c+d x^2\right )^{7/2}}{8 d} \]

Antiderivative was successfully verified.

[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 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 388

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 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^2} \, dx &=-\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 ) \operatorname{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]  time = 0.116308, size = 174, normalized size = 0.8 \[ \sqrt{c+d x^2} \left (\frac{1}{192} x^3 \left (48 a^2 d^2+208 a b c d+59 b^2 c^2\right )+\frac{c x \left (144 a^2 d^2+176 a b c d+5 b^2 c^2\right )}{128 d}-\frac{a^2 c^2}{x}+\frac{1}{48} b d x^5 (16 a d+17 b c)+\frac{1}{8} b^2 d^2 x^7\right )-\frac{5 c^2 \left (-48 a^2 d^2-16 a b c d+b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{128 d^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[c + d*x^2]*(-((a^2*c^2)/x) + (c*(5*b^2*c^2 + 176*a*b*c*d + 144*a^2*d^2)*x)/(128*d) + ((59*b^2*c^2 + 208*a
*b*c*d + 48*a^2*d^2)*x^3)/192 + (b*d*(17*b*c + 16*a*d)*x^5)/48 + (b^2*d^2*x^7)/8) - (5*c^2*(b^2*c^2 - 16*a*b*c
*d - 48*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(128*d^(3/2))

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Maple [A]  time = 0.009, size = 278, normalized size = 1.3 \begin{align*}{\frac{{b}^{2}x}{8\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{{b}^{2}cx}{48\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{b}^{2}{c}^{2}x}{192\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{b}^{2}{c}^{3}x}{128\,d}\sqrt{d{x}^{2}+c}}-{\frac{5\,{b}^{2}{c}^{4}}{128}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{3}{2}}}}+{\frac{abx}{3} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{5\,abcx}{12} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{5\,ab{c}^{2}x}{8}\sqrt{d{x}^{2}+c}}+{\frac{5\,ab{c}^{3}}{8}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){\frac{1}{\sqrt{d}}}}-{\frac{{a}^{2}}{cx} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{{a}^{2}dx}{c} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{a}^{2}dx}{4} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{15\,{a}^{2}cdx}{8}\sqrt{d{x}^{2}+c}}+{\frac{15\,{a}^{2}{c}^{2}}{8}\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^2,x)

[Out]

1/8*b^2*x*(d*x^2+c)^(7/2)/d-1/48*b^2*c/d*x*(d*x^2+c)^(5/2)-5/192*b^2*c^2/d*x*(d*x^2+c)^(3/2)-5/128*b^2*c^3/d*x
*(d*x^2+c)^(1/2)-5/128*b^2*c^4/d^(3/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))+1/3*a*b*x*(d*x^2+c)^(5/2)+5/12*a*b*c*x*(d
*x^2+c)^(3/2)+5/8*a*b*c^2*x*(d*x^2+c)^(1/2)+5/8*a*b*c^3/d^(1/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))-a^2*(d*x^2+c)^(7
/2)/c/x+a^2*d/c*x*(d*x^2+c)^(5/2)+5/4*a^2*d*x*(d*x^2+c)^(3/2)+15/8*a^2*d*c*x*(d*x^2+c)^(1/2)+15/8*a^2*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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.71392, size = 855, normalized size = 3.94 \begin{align*} \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 ] \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^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)]

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Sympy [B]  time = 33.549, size = 496, normalized size = 2.29 \begin{align*} - \frac{a^{2} c^{\frac{5}{2}}}{x \sqrt{1 + \frac{d x^{2}}{c}}} + a^{2} c^{\frac{3}{2}} d x \sqrt{1 + \frac{d x^{2}}{c}} - \frac{7 a^{2} c^{\frac{3}{2}} d x}{8 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 a^{2} \sqrt{c} d^{2} x^{3}}{8 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{15 a^{2} c^{2} \sqrt{d} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8} + \frac{a^{2} d^{3} x^{5}}{4 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} + a b c^{\frac{5}{2}} x \sqrt{1 + \frac{d x^{2}}{c}} + \frac{3 a b c^{\frac{5}{2}} x}{8 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{35 a b c^{\frac{3}{2}} d x^{3}}{24 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{17 a b \sqrt{c} d^{2} x^{5}}{12 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 a b c^{3} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 \sqrt{d}} + \frac{a b d^{3} x^{7}}{3 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 b^{2} c^{\frac{7}{2}} x}{128 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{133 b^{2} c^{\frac{5}{2}} x^{3}}{384 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{127 b^{2} c^{\frac{3}{2}} d x^{5}}{192 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{23 b^{2} \sqrt{c} d^{2} x^{7}}{48 \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{5 b^{2} c^{4} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{128 d^{\frac{3}{2}}} + \frac{b^{2} d^{3} x^{9}}{8 \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**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) - 7*a**2*c**(3/2)*d*x/(8*sqrt(1 +
d*x**2/c)) + 3*a**2*sqrt(c)*d**2*x**3/(8*sqrt(1 + d*x**2/c)) + 15*a**2*c**2*sqrt(d)*asinh(sqrt(d)*x/sqrt(c))/
8 + a**2*d**3*x**5/(4*sqrt(c)*sqrt(1 + d*x**2/c)) + a*b*c**(5/2)*x*sqrt(1 + d*x**2/c) + 3*a*b*c**(5/2)*x/(8*sq
rt(1 + d*x**2/c)) + 35*a*b*c**(3/2)*d*x**3/(24*sqrt(1 + d*x**2/c)) + 17*a*b*sqrt(c)*d**2*x**5/(12*sqrt(1 + d*x
**2/c)) + 5*a*b*c**3*asinh(sqrt(d)*x/sqrt(c))/(8*sqrt(d)) + a*b*d**3*x**7/(3*sqrt(c)*sqrt(1 + d*x**2/c)) + 5*b
**2*c**(7/2)*x/(128*d*sqrt(1 + d*x**2/c)) + 133*b**2*c**(5/2)*x**3/(384*sqrt(1 + d*x**2/c)) + 127*b**2*c**(3/2
)*d*x**5/(192*sqrt(1 + d*x**2/c)) + 23*b**2*sqrt(c)*d**2*x**7/(48*sqrt(1 + d*x**2/c)) - 5*b**2*c**4*asinh(sqrt
(d)*x/sqrt(c))/(128*d**(3/2)) + b**2*d**3*x**9/(8*sqrt(c)*sqrt(1 + d*x**2/c))

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Giac [A]  time = 1.12851, size = 296, normalized size = 1.36 \begin{align*} \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} \sqrt{d} - 16 \, a b c^{3} d^{\frac{3}{2}} - 48 \, a^{2} c^{2} d^{\frac{5}{2}}\right )} \log \left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{256 \, d^{2}} \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^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*sqrt(d) - 16*a*b*c^3*d^(3/2) - 48*a^2*c^2*d^(5/2))*log(
(sqrt(d)*x - sqrt(d*x^2 + c))^2)/d^2