3.364 \(\int \frac{(a+b x^2)^3 (A+B x^2)}{x^{3/2}} \, dx\)

Optimal. Leaf size=83 \[ \frac{2}{3} a^2 x^{3/2} (a B+3 A b)-\frac{2 a^3 A}{\sqrt{x}}+\frac{2}{11} b^2 x^{11/2} (3 a B+A b)+\frac{6}{7} a b x^{7/2} (a B+A b)+\frac{2}{15} b^3 B x^{15/2} \]

[Out]

(-2*a^3*A)/Sqrt[x] + (2*a^2*(3*A*b + a*B)*x^(3/2))/3 + (6*a*b*(A*b + a*B)*x^(7/2))/7 + (2*b^2*(A*b + 3*a*B)*x^
(11/2))/11 + (2*b^3*B*x^(15/2))/15

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Rubi [A]  time = 0.0437627, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {448} \[ \frac{2}{3} a^2 x^{3/2} (a B+3 A b)-\frac{2 a^3 A}{\sqrt{x}}+\frac{2}{11} b^2 x^{11/2} (3 a B+A b)+\frac{6}{7} a b x^{7/2} (a B+A b)+\frac{2}{15} b^3 B x^{15/2} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^3*(A + B*x^2))/x^(3/2),x]

[Out]

(-2*a^3*A)/Sqrt[x] + (2*a^2*(3*A*b + a*B)*x^(3/2))/3 + (6*a*b*(A*b + a*B)*x^(7/2))/7 + (2*b^2*(A*b + 3*a*B)*x^
(11/2))/11 + (2*b^3*B*x^(15/2))/15

Rule 448

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

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^3 \left (A+B x^2\right )}{x^{3/2}} \, dx &=\int \left (\frac{a^3 A}{x^{3/2}}+a^2 (3 A b+a B) \sqrt{x}+3 a b (A b+a B) x^{5/2}+b^2 (A b+3 a B) x^{9/2}+b^3 B x^{13/2}\right ) \, dx\\ &=-\frac{2 a^3 A}{\sqrt{x}}+\frac{2}{3} a^2 (3 A b+a B) x^{3/2}+\frac{6}{7} a b (A b+a B) x^{7/2}+\frac{2}{11} b^2 (A b+3 a B) x^{11/2}+\frac{2}{15} b^3 B x^{15/2}\\ \end{align*}

Mathematica [A]  time = 0.0216523, size = 81, normalized size = 0.98 \[ \frac{330 a^2 b x^2 \left (7 A+3 B x^2\right )-770 a^3 \left (3 A-B x^2\right )+90 a b^2 x^4 \left (11 A+7 B x^2\right )+14 b^3 x^6 \left (15 A+11 B x^2\right )}{1155 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^3*(A + B*x^2))/x^(3/2),x]

[Out]

(-770*a^3*(3*A - B*x^2) + 330*a^2*b*x^2*(7*A + 3*B*x^2) + 90*a*b^2*x^4*(11*A + 7*B*x^2) + 14*b^3*x^6*(15*A + 1
1*B*x^2))/(1155*Sqrt[x])

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Maple [A]  time = 0.005, size = 80, normalized size = 1. \begin{align*} -{\frac{-154\,{b}^{3}B{x}^{8}-210\,{x}^{6}{b}^{3}A-630\,{x}^{6}a{b}^{2}B-990\,{x}^{4}a{b}^{2}A-990\,{x}^{4}{a}^{2}bB-2310\,{x}^{2}A{a}^{2}b-770\,{x}^{2}B{a}^{3}+2310\,{a}^{3}A}{1155}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^3*(B*x^2+A)/x^(3/2),x)

[Out]

-2/1155*(-77*B*b^3*x^8-105*A*b^3*x^6-315*B*a*b^2*x^6-495*A*a*b^2*x^4-495*B*a^2*b*x^4-1155*A*a^2*b*x^2-385*B*a^
3*x^2+1155*A*a^3)/x^(1/2)

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Maxima [A]  time = 1.35472, size = 99, normalized size = 1.19 \begin{align*} \frac{2}{15} \, B b^{3} x^{\frac{15}{2}} + \frac{2}{11} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{11}{2}} + \frac{6}{7} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{7}{2}} - \frac{2 \, A a^{3}}{\sqrt{x}} + \frac{2}{3} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3*(B*x^2+A)/x^(3/2),x, algorithm="maxima")

[Out]

2/15*B*b^3*x^(15/2) + 2/11*(3*B*a*b^2 + A*b^3)*x^(11/2) + 6/7*(B*a^2*b + A*a*b^2)*x^(7/2) - 2*A*a^3/sqrt(x) +
2/3*(B*a^3 + 3*A*a^2*b)*x^(3/2)

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Fricas [A]  time = 0.779691, size = 181, normalized size = 2.18 \begin{align*} \frac{2 \,{\left (77 \, B b^{3} x^{8} + 105 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + 495 \,{\left (B a^{2} b + A a b^{2}\right )} x^{4} - 1155 \, A a^{3} + 385 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}\right )}}{1155 \, \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3*(B*x^2+A)/x^(3/2),x, algorithm="fricas")

[Out]

2/1155*(77*B*b^3*x^8 + 105*(3*B*a*b^2 + A*b^3)*x^6 + 495*(B*a^2*b + A*a*b^2)*x^4 - 1155*A*a^3 + 385*(B*a^3 + 3
*A*a^2*b)*x^2)/sqrt(x)

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Sympy [A]  time = 5.60799, size = 110, normalized size = 1.33 \begin{align*} - \frac{2 A a^{3}}{\sqrt{x}} + 2 A a^{2} b x^{\frac{3}{2}} + \frac{6 A a b^{2} x^{\frac{7}{2}}}{7} + \frac{2 A b^{3} x^{\frac{11}{2}}}{11} + \frac{2 B a^{3} x^{\frac{3}{2}}}{3} + \frac{6 B a^{2} b x^{\frac{7}{2}}}{7} + \frac{6 B a b^{2} x^{\frac{11}{2}}}{11} + \frac{2 B b^{3} x^{\frac{15}{2}}}{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**3*(B*x**2+A)/x**(3/2),x)

[Out]

-2*A*a**3/sqrt(x) + 2*A*a**2*b*x**(3/2) + 6*A*a*b**2*x**(7/2)/7 + 2*A*b**3*x**(11/2)/11 + 2*B*a**3*x**(3/2)/3
+ 6*B*a**2*b*x**(7/2)/7 + 6*B*a*b**2*x**(11/2)/11 + 2*B*b**3*x**(15/2)/15

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Giac [A]  time = 1.33175, size = 104, normalized size = 1.25 \begin{align*} \frac{2}{15} \, B b^{3} x^{\frac{15}{2}} + \frac{6}{11} \, B a b^{2} x^{\frac{11}{2}} + \frac{2}{11} \, A b^{3} x^{\frac{11}{2}} + \frac{6}{7} \, B a^{2} b x^{\frac{7}{2}} + \frac{6}{7} \, A a b^{2} x^{\frac{7}{2}} + \frac{2}{3} \, B a^{3} x^{\frac{3}{2}} + 2 \, A a^{2} b x^{\frac{3}{2}} - \frac{2 \, A a^{3}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3*(B*x^2+A)/x^(3/2),x, algorithm="giac")

[Out]

2/15*B*b^3*x^(15/2) + 6/11*B*a*b^2*x^(11/2) + 2/11*A*b^3*x^(11/2) + 6/7*B*a^2*b*x^(7/2) + 6/7*A*a*b^2*x^(7/2)
+ 2/3*B*a^3*x^(3/2) + 2*A*a^2*b*x^(3/2) - 2*A*a^3/sqrt(x)