3.45 \(\int \sqrt{a+b x^2} (c+d x^2)^3 \, dx\)

Optimal. Leaf size=231 \[ \frac{d x \left (a+b x^2\right )^{3/2} \left (15 a^2 d^2-52 a b c d+72 b^2 c^2\right )}{192 b^3}+\frac{x \sqrt{a+b x^2} \left (24 a^2 b c d^2-5 a^3 d^3-48 a b^2 c^2 d+64 b^3 c^3\right )}{128 b^3}+\frac{a \left (24 a^2 b c d^2-5 a^3 d^3-48 a b^2 c^2 d+64 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{7/2}}+\frac{d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (12 b c-5 a d)}{48 b^2}+\frac{d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b} \]

[Out]

((64*b^3*c^3 - 48*a*b^2*c^2*d + 24*a^2*b*c*d^2 - 5*a^3*d^3)*x*Sqrt[a + b*x^2])/(128*b^3) + (d*(72*b^2*c^2 - 52
*a*b*c*d + 15*a^2*d^2)*x*(a + b*x^2)^(3/2))/(192*b^3) + (d*(12*b*c - 5*a*d)*x*(a + b*x^2)^(3/2)*(c + d*x^2))/(
48*b^2) + (d*x*(a + b*x^2)^(3/2)*(c + d*x^2)^2)/(8*b) + (a*(64*b^3*c^3 - 48*a*b^2*c^2*d + 24*a^2*b*c*d^2 - 5*a
^3*d^3)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(128*b^(7/2))

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Rubi [A]  time = 0.178677, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {416, 528, 388, 195, 217, 206} \[ \frac{d x \left (a+b x^2\right )^{3/2} \left (15 a^2 d^2-52 a b c d+72 b^2 c^2\right )}{192 b^3}+\frac{x \sqrt{a+b x^2} \left (24 a^2 b c d^2-5 a^3 d^3-48 a b^2 c^2 d+64 b^3 c^3\right )}{128 b^3}+\frac{a \left (24 a^2 b c d^2-5 a^3 d^3-48 a b^2 c^2 d+64 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{7/2}}+\frac{d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) (12 b c-5 a d)}{48 b^2}+\frac{d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*x^2]*(c + d*x^2)^3,x]

[Out]

((64*b^3*c^3 - 48*a*b^2*c^2*d + 24*a^2*b*c*d^2 - 5*a^3*d^3)*x*Sqrt[a + b*x^2])/(128*b^3) + (d*(72*b^2*c^2 - 52
*a*b*c*d + 15*a^2*d^2)*x*(a + b*x^2)^(3/2))/(192*b^3) + (d*(12*b*c - 5*a*d)*x*(a + b*x^2)^(3/2)*(c + d*x^2))/(
48*b^2) + (d*x*(a + b*x^2)^(3/2)*(c + d*x^2)^2)/(8*b) + (a*(64*b^3*c^3 - 48*a*b^2*c^2*d + 24*a^2*b*c*d^2 - 5*a
^3*d^3)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(128*b^(7/2))

Rule 416

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^(q - 1))/(b*(n*(p + q) + 1)), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rule 528

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[
(f*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*(n*(p + q + 1) + 1)), x] + Dist[1/(b*(n*(p + q + 1) + 1)), Int[(a +
b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*
d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1
, 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 \sqrt{a+b x^2} \left (c+d x^2\right )^3 \, dx &=\frac{d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}+\frac{\int \sqrt{a+b x^2} \left (c+d x^2\right ) \left (c (8 b c-a d)+d (12 b c-5 a d) x^2\right ) \, dx}{8 b}\\ &=\frac{d (12 b c-5 a d) x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{48 b^2}+\frac{d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}+\frac{\int \sqrt{a+b x^2} \left (c \left (48 b^2 c^2-18 a b c d+5 a^2 d^2\right )+d \left (72 b^2 c^2-52 a b c d+15 a^2 d^2\right ) x^2\right ) \, dx}{48 b^2}\\ &=\frac{d \left (72 b^2 c^2-52 a b c d+15 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{192 b^3}+\frac{d (12 b c-5 a d) x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{48 b^2}+\frac{d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}+\frac{\left (64 b^3 c^3-48 a b^2 c^2 d+24 a^2 b c d^2-5 a^3 d^3\right ) \int \sqrt{a+b x^2} \, dx}{64 b^3}\\ &=\frac{\left (64 b^3 c^3-48 a b^2 c^2 d+24 a^2 b c d^2-5 a^3 d^3\right ) x \sqrt{a+b x^2}}{128 b^3}+\frac{d \left (72 b^2 c^2-52 a b c d+15 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{192 b^3}+\frac{d (12 b c-5 a d) x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{48 b^2}+\frac{d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}+\frac{\left (a \left (64 b^3 c^3-48 a b^2 c^2 d+24 a^2 b c d^2-5 a^3 d^3\right )\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{128 b^3}\\ &=\frac{\left (64 b^3 c^3-48 a b^2 c^2 d+24 a^2 b c d^2-5 a^3 d^3\right ) x \sqrt{a+b x^2}}{128 b^3}+\frac{d \left (72 b^2 c^2-52 a b c d+15 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{192 b^3}+\frac{d (12 b c-5 a d) x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{48 b^2}+\frac{d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}+\frac{\left (a \left (64 b^3 c^3-48 a b^2 c^2 d+24 a^2 b c d^2-5 a^3 d^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{128 b^3}\\ &=\frac{\left (64 b^3 c^3-48 a b^2 c^2 d+24 a^2 b c d^2-5 a^3 d^3\right ) x \sqrt{a+b x^2}}{128 b^3}+\frac{d \left (72 b^2 c^2-52 a b c d+15 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{192 b^3}+\frac{d (12 b c-5 a d) x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{48 b^2}+\frac{d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2}{8 b}+\frac{a \left (64 b^3 c^3-48 a b^2 c^2 d+24 a^2 b c d^2-5 a^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{7/2}}\\ \end{align*}

Mathematica [A]  time = 5.1079, size = 181, normalized size = 0.78 \[ \frac{\sqrt{b} x \sqrt{a+b x^2} \left (-2 a^2 b d^2 \left (36 c+5 d x^2\right )+15 a^3 d^3+8 a b^2 d \left (18 c^2+6 c d x^2+d^2 x^4\right )+48 b^3 \left (6 c^2 d x^2+4 c^3+4 c d^2 x^4+d^3 x^6\right )\right )-3 a \left (-24 a^2 b c d^2+5 a^3 d^3+48 a b^2 c^2 d-64 b^3 c^3\right ) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{384 b^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*x^2]*(c + d*x^2)^3,x]

[Out]

(Sqrt[b]*x*Sqrt[a + b*x^2]*(15*a^3*d^3 - 2*a^2*b*d^2*(36*c + 5*d*x^2) + 8*a*b^2*d*(18*c^2 + 6*c*d*x^2 + d^2*x^
4) + 48*b^3*(4*c^3 + 6*c^2*d*x^2 + 4*c*d^2*x^4 + d^3*x^6)) - 3*a*(-64*b^3*c^3 + 48*a*b^2*c^2*d - 24*a^2*b*c*d^
2 + 5*a^3*d^3)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(384*b^(7/2))

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Maple [A]  time = 0.013, size = 310, normalized size = 1.3 \begin{align*}{\frac{{d}^{3}{x}^{5}}{8\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,a{d}^{3}{x}^{3}}{48\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}{d}^{3}x}{64\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{a}^{3}{d}^{3}x}{128\,{b}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{5\,{d}^{3}{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}}+{\frac{c{d}^{2}{x}^{3}}{2\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,ac{d}^{2}x}{8\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{a}^{2}c{d}^{2}x}{16\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,{a}^{3}c{d}^{2}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}+{\frac{3\,{c}^{2}dx}{4\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,a{c}^{2}dx}{8\,b}\sqrt{b{x}^{2}+a}}-{\frac{3\,{a}^{2}{c}^{2}d}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{{c}^{3}x}{2}\sqrt{b{x}^{2}+a}}+{\frac{{c}^{3}a}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(1/2)*(d*x^2+c)^3,x)

[Out]

1/8*d^3*x^5*(b*x^2+a)^(3/2)/b-5/48*d^3/b^2*a*x^3*(b*x^2+a)^(3/2)+5/64*d^3/b^3*a^2*x*(b*x^2+a)^(3/2)-5/128*d^3/
b^3*a^3*x*(b*x^2+a)^(1/2)-5/128*d^3/b^(7/2)*a^4*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+1/2*c*d^2*x^3*(b*x^2+a)^(3/2)/b-
3/8*c*d^2/b^2*a*x*(b*x^2+a)^(3/2)+3/16*c*d^2/b^2*a^2*x*(b*x^2+a)^(1/2)+3/16*c*d^2/b^(5/2)*a^3*ln(x*b^(1/2)+(b*
x^2+a)^(1/2))+3/4*c^2*d*x*(b*x^2+a)^(3/2)/b-3/8*c^2*d/b*a*x*(b*x^2+a)^(1/2)-3/8*c^2*d/b^(3/2)*a^2*ln(x*b^(1/2)
+(b*x^2+a)^(1/2))+1/2*c^3*x*(b*x^2+a)^(1/2)+1/2*c^3*a/b^(1/2)*ln(x*b^(1/2)+(b*x^2+a)^(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)^(1/2)*(d*x^2+c)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.33364, size = 886, normalized size = 3.84 \begin{align*} \left [-\frac{3 \,{\left (64 \, a b^{3} c^{3} - 48 \, a^{2} b^{2} c^{2} d + 24 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3}\right )} \sqrt{b} \log \left (-2 \, b x^{2} + 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left (48 \, b^{4} d^{3} x^{7} + 8 \,{\left (24 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} x^{5} + 2 \,{\left (144 \, b^{4} c^{2} d + 24 \, a b^{3} c d^{2} - 5 \, a^{2} b^{2} d^{3}\right )} x^{3} + 3 \,{\left (64 \, b^{4} c^{3} + 48 \, a b^{3} c^{2} d - 24 \, a^{2} b^{2} c d^{2} + 5 \, a^{3} b d^{3}\right )} x\right )} \sqrt{b x^{2} + a}}{768 \, b^{4}}, -\frac{3 \,{\left (64 \, a b^{3} c^{3} - 48 \, a^{2} b^{2} c^{2} d + 24 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (48 \, b^{4} d^{3} x^{7} + 8 \,{\left (24 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} x^{5} + 2 \,{\left (144 \, b^{4} c^{2} d + 24 \, a b^{3} c d^{2} - 5 \, a^{2} b^{2} d^{3}\right )} x^{3} + 3 \,{\left (64 \, b^{4} c^{3} + 48 \, a b^{3} c^{2} d - 24 \, a^{2} b^{2} c d^{2} + 5 \, a^{3} b d^{3}\right )} x\right )} \sqrt{b x^{2} + a}}{384 \, b^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^3,x, algorithm="fricas")

[Out]

[-1/768*(3*(64*a*b^3*c^3 - 48*a^2*b^2*c^2*d + 24*a^3*b*c*d^2 - 5*a^4*d^3)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2
+ a)*sqrt(b)*x - a) - 2*(48*b^4*d^3*x^7 + 8*(24*b^4*c*d^2 + a*b^3*d^3)*x^5 + 2*(144*b^4*c^2*d + 24*a*b^3*c*d^2
- 5*a^2*b^2*d^3)*x^3 + 3*(64*b^4*c^3 + 48*a*b^3*c^2*d - 24*a^2*b^2*c*d^2 + 5*a^3*b*d^3)*x)*sqrt(b*x^2 + a))/b
^4, -1/384*(3*(64*a*b^3*c^3 - 48*a^2*b^2*c^2*d + 24*a^3*b*c*d^2 - 5*a^4*d^3)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b
*x^2 + a)) - (48*b^4*d^3*x^7 + 8*(24*b^4*c*d^2 + a*b^3*d^3)*x^5 + 2*(144*b^4*c^2*d + 24*a*b^3*c*d^2 - 5*a^2*b^
2*d^3)*x^3 + 3*(64*b^4*c^3 + 48*a*b^3*c^2*d - 24*a^2*b^2*c*d^2 + 5*a^3*b*d^3)*x)*sqrt(b*x^2 + a))/b^4]

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Sympy [B]  time = 17.9835, size = 484, normalized size = 2.1 \begin{align*} \frac{5 a^{\frac{7}{2}} d^{3} x}{128 b^{3} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{3 a^{\frac{5}{2}} c d^{2} x}{16 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 a^{\frac{5}{2}} d^{3} x^{3}}{384 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 a^{\frac{3}{2}} c^{2} d x}{8 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{a^{\frac{3}{2}} c d^{2} x^{3}}{16 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{a^{\frac{3}{2}} d^{3} x^{5}}{192 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{\sqrt{a} c^{3} x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{9 \sqrt{a} c^{2} d x^{3}}{8 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 \sqrt{a} c d^{2} x^{5}}{8 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{7 \sqrt{a} d^{3} x^{7}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{5 a^{4} d^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128 b^{\frac{7}{2}}} + \frac{3 a^{3} c d^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 b^{\frac{5}{2}}} - \frac{3 a^{2} c^{2} d \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{3}{2}}} + \frac{a c^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 \sqrt{b}} + \frac{3 b c^{2} d x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{b c d^{2} x^{7}}{2 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{b d^{3} x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(1/2)*(d*x**2+c)**3,x)

[Out]

5*a**(7/2)*d**3*x/(128*b**3*sqrt(1 + b*x**2/a)) - 3*a**(5/2)*c*d**2*x/(16*b**2*sqrt(1 + b*x**2/a)) + 5*a**(5/2
)*d**3*x**3/(384*b**2*sqrt(1 + b*x**2/a)) + 3*a**(3/2)*c**2*d*x/(8*b*sqrt(1 + b*x**2/a)) - a**(3/2)*c*d**2*x**
3/(16*b*sqrt(1 + b*x**2/a)) - a**(3/2)*d**3*x**5/(192*b*sqrt(1 + b*x**2/a)) + sqrt(a)*c**3*x*sqrt(1 + b*x**2/a
)/2 + 9*sqrt(a)*c**2*d*x**3/(8*sqrt(1 + b*x**2/a)) + 5*sqrt(a)*c*d**2*x**5/(8*sqrt(1 + b*x**2/a)) + 7*sqrt(a)*
d**3*x**7/(48*sqrt(1 + b*x**2/a)) - 5*a**4*d**3*asinh(sqrt(b)*x/sqrt(a))/(128*b**(7/2)) + 3*a**3*c*d**2*asinh(
sqrt(b)*x/sqrt(a))/(16*b**(5/2)) - 3*a**2*c**2*d*asinh(sqrt(b)*x/sqrt(a))/(8*b**(3/2)) + a*c**3*asinh(sqrt(b)*
x/sqrt(a))/(2*sqrt(b)) + 3*b*c**2*d*x**5/(4*sqrt(a)*sqrt(1 + b*x**2/a)) + b*c*d**2*x**7/(2*sqrt(a)*sqrt(1 + b*
x**2/a)) + b*d**3*x**9/(8*sqrt(a)*sqrt(1 + b*x**2/a))

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Giac [A]  time = 1.80805, size = 271, normalized size = 1.17 \begin{align*} \frac{1}{384} \,{\left (2 \,{\left (4 \,{\left (6 \, d^{3} x^{2} + \frac{24 \, b^{6} c d^{2} + a b^{5} d^{3}}{b^{6}}\right )} x^{2} + \frac{144 \, b^{6} c^{2} d + 24 \, a b^{5} c d^{2} - 5 \, a^{2} b^{4} d^{3}}{b^{6}}\right )} x^{2} + \frac{3 \,{\left (64 \, b^{6} c^{3} + 48 \, a b^{5} c^{2} d - 24 \, a^{2} b^{4} c d^{2} + 5 \, a^{3} b^{3} d^{3}\right )}}{b^{6}}\right )} \sqrt{b x^{2} + a} x - \frac{{\left (64 \, a b^{3} c^{3} - 48 \, a^{2} b^{2} c^{2} d + 24 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3}\right )} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{128 \, b^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^3,x, algorithm="giac")

[Out]

1/384*(2*(4*(6*d^3*x^2 + (24*b^6*c*d^2 + a*b^5*d^3)/b^6)*x^2 + (144*b^6*c^2*d + 24*a*b^5*c*d^2 - 5*a^2*b^4*d^3
)/b^6)*x^2 + 3*(64*b^6*c^3 + 48*a*b^5*c^2*d - 24*a^2*b^4*c*d^2 + 5*a^3*b^3*d^3)/b^6)*sqrt(b*x^2 + a)*x - 1/128
*(64*a*b^3*c^3 - 48*a^2*b^2*c^2*d + 24*a^3*b*c*d^2 - 5*a^4*d^3)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(7/2)