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

Optimal. Leaf size=74 \[ -\frac {2 b \left (c+d x^2\right )^{3/2} (b c-a d)}{3 d^3}+\frac {\sqrt {c+d x^2} (b c-a d)^2}{d^3}+\frac {b^2 \left (c+d x^2\right )^{5/2}}{5 d^3} \]

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Rubi [A]  time = 0.05, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {444, 43} \begin {gather*} -\frac {2 b \left (c+d x^2\right )^{3/2} (b c-a d)}{3 d^3}+\frac {\sqrt {c+d x^2} (b c-a d)^2}{d^3}+\frac {b^2 \left (c+d x^2\right )^{5/2}}{5 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*c - a*d)^2*Sqrt[c + d*x^2])/d^3 - (2*b*(b*c - a*d)*(c + d*x^2)^(3/2))/(3*d^3) + (b^2*(c + d*x^2)^(5/2))/(5
*d^3)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 66, normalized size = 0.89 \begin {gather*} \frac {\sqrt {c+d x^2} \left (15 a^2 d^2+10 a b d \left (d x^2-2 c\right )+b^2 \left (8 c^2-4 c d x^2+3 d^2 x^4\right )\right )}{15 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + d*x^2]*(15*a^2*d^2 + 10*a*b*d*(-2*c + d*x^2) + b^2*(8*c^2 - 4*c*d*x^2 + 3*d^2*x^4)))/(15*d^3)

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IntegrateAlgebraic [A]  time = 0.04, size = 72, normalized size = 0.97 \begin {gather*} \frac {\sqrt {c+d x^2} \left (15 a^2 d^2-20 a b c d+10 a b d^2 x^2+8 b^2 c^2-4 b^2 c d x^2+3 b^2 d^2 x^4\right )}{15 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x*(a + b*x^2)^2)/Sqrt[c + d*x^2],x]

[Out]

(Sqrt[c + d*x^2]*(8*b^2*c^2 - 20*a*b*c*d + 15*a^2*d^2 - 4*b^2*c*d*x^2 + 10*a*b*d^2*x^2 + 3*b^2*d^2*x^4))/(15*d
^3)

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fricas [A]  time = 1.48, size = 68, normalized size = 0.92 \begin {gather*} \frac {{\left (3 \, b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 20 \, a b c d + 15 \, a^{2} d^{2} - 2 \, {\left (2 \, b^{2} c d - 5 \, a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{15 \, d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(3*b^2*d^2*x^4 + 8*b^2*c^2 - 20*a*b*c*d + 15*a^2*d^2 - 2*(2*b^2*c*d - 5*a*b*d^2)*x^2)*sqrt(d*x^2 + c)/d^3

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giac [A]  time = 0.33, size = 84, normalized size = 1.14 \begin {gather*} \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {d x^{2} + c}}{d^{3}} + \frac {3 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} - 10 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c + 10 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d}{15 \, d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(d*x^2 + c)/d^3 + 1/15*(3*(d*x^2 + c)^(5/2)*b^2 - 10*(d*x^2 + c)^(3/2)*b^2
*c + 10*(d*x^2 + c)^(3/2)*a*b*d)/d^3

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maple [A]  time = 0.01, size = 69, normalized size = 0.93 \begin {gather*} \frac {\sqrt {d \,x^{2}+c}\, \left (3 b^{2} x^{4} d^{2}+10 a b \,d^{2} x^{2}-4 b^{2} c d \,x^{2}+15 a^{2} d^{2}-20 a b c d +8 b^{2} c^{2}\right )}{15 d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(d*x^2+c)^(1/2)*(3*b^2*d^2*x^4+10*a*b*d^2*x^2-4*b^2*c*d*x^2+15*a^2*d^2-20*a*b*c*d+8*b^2*c^2)/d^3

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maxima [A]  time = 0.89, size = 114, normalized size = 1.54 \begin {gather*} \frac {\sqrt {d x^{2} + c} b^{2} x^{4}}{5 \, d} - \frac {4 \, \sqrt {d x^{2} + c} b^{2} c x^{2}}{15 \, d^{2}} + \frac {2 \, \sqrt {d x^{2} + c} a b x^{2}}{3 \, d} + \frac {8 \, \sqrt {d x^{2} + c} b^{2} c^{2}}{15 \, d^{3}} - \frac {4 \, \sqrt {d x^{2} + c} a b c}{3 \, d^{2}} + \frac {\sqrt {d x^{2} + c} a^{2}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*sqrt(d*x^2 + c)*b^2*x^4/d - 4/15*sqrt(d*x^2 + c)*b^2*c*x^2/d^2 + 2/3*sqrt(d*x^2 + c)*a*b*x^2/d + 8/15*sqrt
(d*x^2 + c)*b^2*c^2/d^3 - 4/3*sqrt(d*x^2 + c)*a*b*c/d^2 + sqrt(d*x^2 + c)*a^2/d

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mupad [B]  time = 0.65, size = 68, normalized size = 0.92 \begin {gather*} \sqrt {d\,x^2+c}\,\left (\frac {15\,a^2\,d^2-20\,a\,b\,c\,d+8\,b^2\,c^2}{15\,d^3}+\frac {b^2\,x^4}{5\,d}+\frac {2\,b\,x^2\,\left (5\,a\,d-2\,b\,c\right )}{15\,d^2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c + d*x^2)^(1/2)*((15*a^2*d^2 + 8*b^2*c^2 - 20*a*b*c*d)/(15*d^3) + (b^2*x^4)/(5*d) + (2*b*x^2*(5*a*d - 2*b*c)
)/(15*d^2))

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sympy [A]  time = 1.88, size = 158, normalized size = 2.14 \begin {gather*} \begin {cases} \frac {a^{2} \sqrt {c + d x^{2}}}{d} - \frac {4 a b c \sqrt {c + d x^{2}}}{3 d^{2}} + \frac {2 a b x^{2} \sqrt {c + d x^{2}}}{3 d} + \frac {8 b^{2} c^{2} \sqrt {c + d x^{2}}}{15 d^{3}} - \frac {4 b^{2} c x^{2} \sqrt {c + d x^{2}}}{15 d^{2}} + \frac {b^{2} x^{4} \sqrt {c + d x^{2}}}{5 d} & \text {for}\: d \neq 0 \\\frac {\frac {a^{2} x^{2}}{2} + \frac {a b x^{4}}{2} + \frac {b^{2} x^{6}}{6}}{\sqrt {c}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*sqrt(c + d*x**2)/d - 4*a*b*c*sqrt(c + d*x**2)/(3*d**2) + 2*a*b*x**2*sqrt(c + d*x**2)/(3*d) + 8
*b**2*c**2*sqrt(c + d*x**2)/(15*d**3) - 4*b**2*c*x**2*sqrt(c + d*x**2)/(15*d**2) + b**2*x**4*sqrt(c + d*x**2)/
(5*d), Ne(d, 0)), ((a**2*x**2/2 + a*b*x**4/2 + b**2*x**6/6)/sqrt(c), True))

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