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

Optimal. Leaf size=61 \[ -\frac {2 a^2 A}{\sqrt {x}}+\frac {2}{7} b x^{7/2} (2 a B+A b)+\frac {2}{3} a x^{3/2} (a B+2 A b)+\frac {2}{11} b^2 B x^{11/2} \]

[Out]

2/3*a*(2*A*b+B*a)*x^(3/2)+2/7*b*(A*b+2*B*a)*x^(7/2)+2/11*b^2*B*x^(11/2)-2*a^2*A/x^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, 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} \[ -\frac {2 a^2 A}{\sqrt {x}}+\frac {2}{7} b x^{7/2} (2 a B+A b)+\frac {2}{3} a x^{3/2} (a B+2 A b)+\frac {2}{11} b^2 B x^{11/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.02, size = 60, normalized size = 0.98 \[ \frac {-154 a^2 \left (3 A-B x^2\right )+44 a b x^2 \left (7 A+3 B x^2\right )+6 b^2 x^4 \left (11 A+7 B x^2\right )}{231 \sqrt {x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-154*a^2*(3*A - B*x^2) + 44*a*b*x^2*(7*A + 3*B*x^2) + 6*b^2*x^4*(11*A + 7*B*x^2))/(231*Sqrt[x])

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fricas [A]  time = 0.42, size = 53, normalized size = 0.87 \[ \frac {2 \, {\left (21 \, B b^{2} x^{6} + 33 \, {\left (2 \, B a b + A b^{2}\right )} x^{4} - 231 \, A a^{2} + 77 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )}}{231 \, \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/231*(21*B*b^2*x^6 + 33*(2*B*a*b + A*b^2)*x^4 - 231*A*a^2 + 77*(B*a^2 + 2*A*a*b)*x^2)/sqrt(x)

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giac [A]  time = 0.36, size = 53, normalized size = 0.87 \[ \frac {2}{11} \, B b^{2} x^{\frac {11}{2}} + \frac {4}{7} \, B a b x^{\frac {7}{2}} + \frac {2}{7} \, A b^{2} x^{\frac {7}{2}} + \frac {2}{3} \, B a^{2} x^{\frac {3}{2}} + \frac {4}{3} \, A a b x^{\frac {3}{2}} - \frac {2 \, A a^{2}}{\sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*B*b^2*x^(11/2) + 4/7*B*a*b*x^(7/2) + 2/7*A*b^2*x^(7/2) + 2/3*B*a^2*x^(3/2) + 4/3*A*a*b*x^(3/2) - 2*A*a^2/
sqrt(x)

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maple [A]  time = 0.01, size = 56, normalized size = 0.92 \[ -\frac {2 \left (-21 B \,b^{2} x^{6}-33 A \,b^{2} x^{4}-66 B a b \,x^{4}-154 A a b \,x^{2}-77 B \,a^{2} x^{2}+231 a^{2} A \right )}{231 \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/231*(-21*B*b^2*x^6-33*A*b^2*x^4-66*B*a*b*x^4-154*A*a*b*x^2-77*B*a^2*x^2+231*A*a^2)/x^(1/2)

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maxima [A]  time = 1.03, size = 51, normalized size = 0.84 \[ \frac {2}{11} \, B b^{2} x^{\frac {11}{2}} + \frac {2}{7} \, {\left (2 \, B a b + A b^{2}\right )} x^{\frac {7}{2}} - \frac {2 \, A a^{2}}{\sqrt {x}} + \frac {2}{3} \, {\left (B a^{2} + 2 \, A a b\right )} x^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.05, size = 51, normalized size = 0.84 \[ x^{3/2}\,\left (\frac {2\,B\,a^2}{3}+\frac {4\,A\,b\,a}{3}\right )+x^{7/2}\,\left (\frac {2\,A\,b^2}{7}+\frac {4\,B\,a\,b}{7}\right )-\frac {2\,A\,a^2}{\sqrt {x}}+\frac {2\,B\,b^2\,x^{11/2}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(3/2)*((2*B*a^2)/3 + (4*A*a*b)/3) + x^(7/2)*((2*A*b^2)/7 + (4*B*a*b)/7) - (2*A*a^2)/x^(1/2) + (2*B*b^2*x^(11
/2))/11

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sympy [A]  time = 2.42, size = 78, normalized size = 1.28 \[ - \frac {2 A a^{2}}{\sqrt {x}} + \frac {4 A a b x^{\frac {3}{2}}}{3} + \frac {2 A b^{2} x^{\frac {7}{2}}}{7} + \frac {2 B a^{2} x^{\frac {3}{2}}}{3} + \frac {4 B a b x^{\frac {7}{2}}}{7} + \frac {2 B b^{2} x^{\frac {11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*A*a**2/sqrt(x) + 4*A*a*b*x**(3/2)/3 + 2*A*b**2*x**(7/2)/7 + 2*B*a**2*x**(3/2)/3 + 4*B*a*b*x**(7/2)/7 + 2*B*
b**2*x**(11/2)/11

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