3.542 \(\int x (a+b x^2)^{5/2} (A+B x^2) \, dx\)

Optimal. Leaf size=46 \[ \frac{\left (a+b x^2\right )^{7/2} (A b-a B)}{7 b^2}+\frac{B \left (a+b x^2\right )^{9/2}}{9 b^2} \]

[Out]

((A*b - a*B)*(a + b*x^2)^(7/2))/(7*b^2) + (B*(a + b*x^2)^(9/2))/(9*b^2)

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Rubi [A]  time = 0.0345383, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {444, 43} \[ \frac{\left (a+b x^2\right )^{7/2} (A b-a B)}{7 b^2}+\frac{B \left (a+b x^2\right )^{9/2}}{9 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2)^(5/2)*(A + B*x^2),x]

[Out]

((A*b - a*B)*(a + b*x^2)^(7/2))/(7*b^2) + (B*(a + b*x^2)^(9/2))/(9*b^2)

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]

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])

Rubi steps

\begin{align*} \int x \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int (a+b x)^{5/2} (A+B x) \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{(A b-a B) (a+b x)^{5/2}}{b}+\frac{B (a+b x)^{7/2}}{b}\right ) \, dx,x,x^2\right )\\ &=\frac{(A b-a B) \left (a+b x^2\right )^{7/2}}{7 b^2}+\frac{B \left (a+b x^2\right )^{9/2}}{9 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0258787, size = 34, normalized size = 0.74 \[ \frac{\left (a+b x^2\right )^{7/2} \left (-2 a B+9 A b+7 b B x^2\right )}{63 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^2)^(5/2)*(A + B*x^2),x]

[Out]

((a + b*x^2)^(7/2)*(9*A*b - 2*a*B + 7*b*B*x^2))/(63*b^2)

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Maple [A]  time = 0.005, size = 31, normalized size = 0.7 \begin{align*}{\frac{7\,bB{x}^{2}+9\,Ab-2\,Ba}{63\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x^2+a)^(5/2)*(B*x^2+A),x)

[Out]

1/63*(b*x^2+a)^(7/2)*(7*B*b*x^2+9*A*b-2*B*a)/b^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(x*(b*x^2+a)^(5/2)*(B*x^2+A),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.62784, size = 211, normalized size = 4.59 \begin{align*} \frac{{\left (7 \, B b^{4} x^{8} +{\left (19 \, B a b^{3} + 9 \, A b^{4}\right )} x^{6} - 2 \, B a^{4} + 9 \, A a^{3} b + 3 \,{\left (5 \, B a^{2} b^{2} + 9 \, A a b^{3}\right )} x^{4} +{\left (B a^{3} b + 27 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{63 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^(5/2)*(B*x^2+A),x, algorithm="fricas")

[Out]

1/63*(7*B*b^4*x^8 + (19*B*a*b^3 + 9*A*b^4)*x^6 - 2*B*a^4 + 9*A*a^3*b + 3*(5*B*a^2*b^2 + 9*A*a*b^3)*x^4 + (B*a^
3*b + 27*A*a^2*b^2)*x^2)*sqrt(b*x^2 + a)/b^2

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Sympy [A]  time = 3.57975, size = 209, normalized size = 4.54 \begin{align*} \begin{cases} \frac{A a^{3} \sqrt{a + b x^{2}}}{7 b} + \frac{3 A a^{2} x^{2} \sqrt{a + b x^{2}}}{7} + \frac{3 A a b x^{4} \sqrt{a + b x^{2}}}{7} + \frac{A b^{2} x^{6} \sqrt{a + b x^{2}}}{7} - \frac{2 B a^{4} \sqrt{a + b x^{2}}}{63 b^{2}} + \frac{B a^{3} x^{2} \sqrt{a + b x^{2}}}{63 b} + \frac{5 B a^{2} x^{4} \sqrt{a + b x^{2}}}{21} + \frac{19 B a b x^{6} \sqrt{a + b x^{2}}}{63} + \frac{B b^{2} x^{8} \sqrt{a + b x^{2}}}{9} & \text{for}\: b \neq 0 \\a^{\frac{5}{2}} \left (\frac{A x^{2}}{2} + \frac{B x^{4}}{4}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x**2+a)**(5/2)*(B*x**2+A),x)

[Out]

Piecewise((A*a**3*sqrt(a + b*x**2)/(7*b) + 3*A*a**2*x**2*sqrt(a + b*x**2)/7 + 3*A*a*b*x**4*sqrt(a + b*x**2)/7
+ A*b**2*x**6*sqrt(a + b*x**2)/7 - 2*B*a**4*sqrt(a + b*x**2)/(63*b**2) + B*a**3*x**2*sqrt(a + b*x**2)/(63*b) +
5*B*a**2*x**4*sqrt(a + b*x**2)/21 + 19*B*a*b*x**6*sqrt(a + b*x**2)/63 + B*b**2*x**8*sqrt(a + b*x**2)/9, Ne(b,
0)), (a**(5/2)*(A*x**2/2 + B*x**4/4), True))

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Giac [B]  time = 1.09716, size = 304, normalized size = 6.61 \begin{align*} \frac{105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a^{2} + 42 \,{\left (3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a\right )} A a + \frac{21 \,{\left (3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a\right )} B a^{2}}{b} + 3 \,{\left (15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2}\right )} A + \frac{6 \,{\left (15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2}\right )} B a}{b} + \frac{{\left (35 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3}\right )} B}{b}}{315 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^(5/2)*(B*x^2+A),x, algorithm="giac")

[Out]

1/315*(105*(b*x^2 + a)^(3/2)*A*a^2 + 42*(3*(b*x^2 + a)^(5/2) - 5*(b*x^2 + a)^(3/2)*a)*A*a + 21*(3*(b*x^2 + a)^
(5/2) - 5*(b*x^2 + a)^(3/2)*a)*B*a^2/b + 3*(15*(b*x^2 + a)^(7/2) - 42*(b*x^2 + a)^(5/2)*a + 35*(b*x^2 + a)^(3/
2)*a^2)*A + 6*(15*(b*x^2 + a)^(7/2) - 42*(b*x^2 + a)^(5/2)*a + 35*(b*x^2 + a)^(3/2)*a^2)*B*a/b + (35*(b*x^2 +
a)^(9/2) - 135*(b*x^2 + a)^(7/2)*a + 189*(b*x^2 + a)^(5/2)*a^2 - 105*(b*x^2 + a)^(3/2)*a^3)*B/b)/b