3.1.45 \(\int \sqrt {a+b x^2} (c+d x^2)^3 \, dx\) [45]

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

[Out]

1/192*d*(15*a^2*d^2-52*a*b*c*d+72*b^2*c^2)*x*(b*x^2+a)^(3/2)/b^3+1/48*d*(-5*a*d+12*b*c)*x*(b*x^2+a)^(3/2)*(d*x
^2+c)/b^2+1/8*d*x*(b*x^2+a)^(3/2)*(d*x^2+c)^2/b+1/128*a*(-5*a^3*d^3+24*a^2*b*c*d^2-48*a*b^2*c^2*d+64*b^3*c^3)*
arctanh(x*b^(1/2)/(b*x^2+a)^(1/2))/b^(7/2)+1/128*(-5*a^3*d^3+24*a^2*b*c*d^2-48*a*b^2*c^2*d+64*b^3*c^3)*x*(b*x^
2+a)^(1/2)/b^3

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Rubi [A]
time = 0.12, antiderivative size = 231, normalized size of antiderivative = 1.00, 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 = {427, 542, 396, 201, 223, 212} \begin {gather*} \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 (-5 a^3 d^3+24 a^2 b c d^2-48 a b^2 c^2 d+64 b^3 c^3\right )}{128 b^3}+\frac {a \left (-5 a^3 d^3+24 a^2 b c d^2-48 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} \end {gather*}

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 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 427

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 542

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]

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 ) \text {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*}

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Mathematica [A]
time = 0.25, size = 180, normalized size = 0.78 \begin {gather*} \frac {\sqrt {b} x \sqrt {a+b x^2} \left (15 a^3 d^3-2 a^2 b d^2 \left (36 c+5 d x^2\right )+8 a b^2 d \left (18 c^2+6 c d x^2+d^2 x^4\right )+48 b^3 \left (4 c^3+6 c^2 d x^2+4 c d^2 x^4+d^3 x^6\right )\right )+3 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 ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{384 b^{7/2}} \end {gather*}

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[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(384*b^(7/2))

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Maple [A]
time = 0.07, size = 300, normalized size = 1.30

method	result	size
risch	\(\frac {x \left (48 b^{3} d^{3} x^{6}+8 a \,b^{2} d^{3} x^{4}+192 b^{3} c \,d^{2} x^{4}-10 a^{2} b \,d^{3} x^{2}+48 a \,b^{2} c \,d^{2} x^{2}+288 b^{3} c^{2} d \,x^{2}+15 a^{3} d^{3}-72 a^{2} b c \,d^{2}+144 a \,b^{2} c^{2} d +192 b^{3} c^{3}\right ) \sqrt {b \,x^{2}+a}}{384 b^{3}}-\frac {5 a^{4} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) d^{3}}{128 b^{\frac {7}{2}}}+\frac {3 a^{3} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) c \,d^{2}}{16 b^{\frac {5}{2}}}-\frac {3 a^{2} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) c^{2} d}{8 b^{\frac {3}{2}}}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) c^{3}}{2 \sqrt {b}}\)	\(234\)
default	\(d^{3} \left (\frac {x^{5} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{8 b}-\frac {5 a \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 b}-\frac {a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4 b}\right )}{2 b}\right )}{8 b}\right )+3 c \,d^{2} \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 b}-\frac {a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4 b}\right )}{2 b}\right )+3 c^{2} d \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 b}-\frac {a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4 b}\right )+c^{3} \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )\)	\(300\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^3*(1/8*x^5*(b*x^2+a)^(3/2)/b-5/8*a/b*(1/6*x^3*(b*x^2+a)^(3/2)/b-1/2*a/b*(1/4*x*(b*x^2+a)^(3/2)/b-1/4*a/b*(1/
2*x*(b*x^2+a)^(1/2)+1/2*a/b^(1/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))))))+3*c*d^2*(1/6*x^3*(b*x^2+a)^(3/2)/b-1/2*a/b
*(1/4*x*(b*x^2+a)^(3/2)/b-1/4*a/b*(1/2*x*(b*x^2+a)^(1/2)+1/2*a/b^(1/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2)))))+3*c^2*
d*(1/4*x*(b*x^2+a)^(3/2)/b-1/4*a/b*(1/2*x*(b*x^2+a)^(1/2)+1/2*a/b^(1/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))))+c^3*(1
/2*x*(b*x^2+a)^(1/2)+1/2*a/b^(1/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2)))

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Maxima [A]
time = 0.30, size = 281, normalized size = 1.22 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} d^{3} x^{5}}{8 \, b} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} c d^{2} x^{3}}{2 \, b} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a d^{3} x^{3}}{48 \, b^{2}} + \frac {1}{2} \, \sqrt {b x^{2} + a} c^{3} x + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} c^{2} d x}{4 \, b} - \frac {3 \, \sqrt {b x^{2} + a} a c^{2} d x}{8 \, b} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a c d^{2} x}{8 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} a^{2} c d^{2} x}{16 \, b^{2}} + \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} d^{3} x}{64 \, b^{3}} - \frac {5 \, \sqrt {b x^{2} + a} a^{3} d^{3} x}{128 \, b^{3}} + \frac {a c^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {b}} - \frac {3 \, a^{2} c^{2} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {3}{2}}} + \frac {3 \, a^{3} c d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {5}{2}}} - \frac {5 \, a^{4} d^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {7}{2}}} \end {gather*}

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]

1/8*(b*x^2 + a)^(3/2)*d^3*x^5/b + 1/2*(b*x^2 + a)^(3/2)*c*d^2*x^3/b - 5/48*(b*x^2 + a)^(3/2)*a*d^3*x^3/b^2 + 1
/2*sqrt(b*x^2 + a)*c^3*x + 3/4*(b*x^2 + a)^(3/2)*c^2*d*x/b - 3/8*sqrt(b*x^2 + a)*a*c^2*d*x/b - 3/8*(b*x^2 + a)
^(3/2)*a*c*d^2*x/b^2 + 3/16*sqrt(b*x^2 + a)*a^2*c*d^2*x/b^2 + 5/64*(b*x^2 + a)^(3/2)*a^2*d^3*x/b^3 - 5/128*sqr
t(b*x^2 + a)*a^3*d^3*x/b^3 + 1/2*a*c^3*arcsinh(b*x/sqrt(a*b))/sqrt(b) - 3/8*a^2*c^2*d*arcsinh(b*x/sqrt(a*b))/b
^(3/2) + 3/16*a^3*c*d^2*arcsinh(b*x/sqrt(a*b))/b^(5/2) - 5/128*a^4*d^3*arcsinh(b*x/sqrt(a*b))/b^(7/2)

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Fricas [A]
time = 2.27, size = 398, normalized size = 1.72 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 484 vs. \(2 (233) = 466\).
time = 28.78, size = 484, normalized size = 2.10 \begin {gather*} \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 {gather*}

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 = 0.96, size = 201, normalized size = 0.87 \begin {gather*} \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 {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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