3.4.40 \(\int \frac {(a+b x^2)^2 (A+B x^2)}{x^{7/2}} \, dx\)

Optimal. Leaf size=61 \[ -\frac {2 a^2 A}{5 x^{5/2}}+\frac {2}{3} b x^{3/2} (2 a B+A b)-\frac {2 a (a B+2 A b)}{\sqrt {x}}+\frac {2}{7} b^2 B x^{7/2} \]

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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} \begin {gather*} -\frac {2 a^2 A}{5 x^{5/2}}+\frac {2}{3} b x^{3/2} (2 a B+A b)-\frac {2 a (a B+2 A b)}{\sqrt {x}}+\frac {2}{7} b^2 B x^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^2*(A + B*x^2))/x^(7/2),x]

[Out]

(-2*a^2*A)/(5*x^(5/2)) - (2*a*(2*A*b + a*B))/Sqrt[x] + (2*b*(A*b + 2*a*B)*x^(3/2))/3 + (2*b^2*B*x^(7/2))/7

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

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Mathematica [A]  time = 0.02, size = 57, normalized size = 0.93 \begin {gather*} \frac {-42 a^2 \left (A+5 B x^2\right )+140 a b x^2 \left (B x^2-3 A\right )+10 b^2 x^4 \left (7 A+3 B x^2\right )}{105 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^2*(A + B*x^2))/x^(7/2),x]

[Out]

(140*a*b*x^2*(-3*A + B*x^2) + 10*b^2*x^4*(7*A + 3*B*x^2) - 42*a^2*(A + 5*B*x^2))/(105*x^(5/2))

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IntegrateAlgebraic [A]  time = 0.04, size = 59, normalized size = 0.97 \begin {gather*} \frac {2 \left (-21 a^2 A-105 a^2 B x^2-210 a A b x^2+70 a b B x^4+35 A b^2 x^4+15 b^2 B x^6\right )}{105 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b*x^2)^2*(A + B*x^2))/x^(7/2),x]

[Out]

(2*(-21*a^2*A - 210*a*A*b*x^2 - 105*a^2*B*x^2 + 35*A*b^2*x^4 + 70*a*b*B*x^4 + 15*b^2*B*x^6))/(105*x^(5/2))

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fricas [A]  time = 1.05, size = 53, normalized size = 0.87 \begin {gather*} \frac {2 \, {\left (15 \, B b^{2} x^{6} + 35 \, {\left (2 \, B a b + A b^{2}\right )} x^{4} - 21 \, A a^{2} - 105 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )}}{105 \, x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(B*x^2+A)/x^(7/2),x, algorithm="fricas")

[Out]

2/105*(15*B*b^2*x^6 + 35*(2*B*a*b + A*b^2)*x^4 - 21*A*a^2 - 105*(B*a^2 + 2*A*a*b)*x^2)/x^(5/2)

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giac [A]  time = 0.40, size = 55, normalized size = 0.90 \begin {gather*} \frac {2}{7} \, B b^{2} x^{\frac {7}{2}} + \frac {4}{3} \, B a b x^{\frac {3}{2}} + \frac {2}{3} \, A b^{2} x^{\frac {3}{2}} - \frac {2 \, {\left (5 \, B a^{2} x^{2} + 10 \, A a b x^{2} + A a^{2}\right )}}{5 \, x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(B*x^2+A)/x^(7/2),x, algorithm="giac")

[Out]

2/7*B*b^2*x^(7/2) + 4/3*B*a*b*x^(3/2) + 2/3*A*b^2*x^(3/2) - 2/5*(5*B*a^2*x^2 + 10*A*a*b*x^2 + A*a^2)/x^(5/2)

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maple [A]  time = 0.01, size = 56, normalized size = 0.92 \begin {gather*} -\frac {2 \left (-15 B \,b^{2} x^{6}-35 A \,b^{2} x^{4}-70 B a b \,x^{4}+210 A a b \,x^{2}+105 B \,a^{2} x^{2}+21 a^{2} A \right )}{105 x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2*(B*x^2+A)/x^(7/2),x)

[Out]

-2/105*(-15*B*b^2*x^6-35*A*b^2*x^4-70*B*a*b*x^4+210*A*a*b*x^2+105*B*a^2*x^2+21*A*a^2)/x^(5/2)

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maxima [A]  time = 0.99, size = 53, normalized size = 0.87 \begin {gather*} \frac {2}{7} \, B b^{2} x^{\frac {7}{2}} + \frac {2}{3} \, {\left (2 \, B a b + A b^{2}\right )} x^{\frac {3}{2}} - \frac {2 \, {\left (A a^{2} + 5 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )}}{5 \, x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(B*x^2+A)/x^(7/2),x, algorithm="maxima")

[Out]

2/7*B*b^2*x^(7/2) + 2/3*(2*B*a*b + A*b^2)*x^(3/2) - 2/5*(A*a^2 + 5*(B*a^2 + 2*A*a*b)*x^2)/x^(5/2)

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mupad [B]  time = 0.25, size = 55, normalized size = 0.90 \begin {gather*} -\frac {210\,B\,a^2\,x^2+42\,A\,a^2-140\,B\,a\,b\,x^4+420\,A\,a\,b\,x^2-30\,B\,b^2\,x^6-70\,A\,b^2\,x^4}{105\,x^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(a + b*x^2)^2)/x^(7/2),x)

[Out]

-(42*A*a^2 + 210*B*a^2*x^2 - 70*A*b^2*x^4 - 30*B*b^2*x^6 + 420*A*a*b*x^2 - 140*B*a*b*x^4)/(105*x^(5/2))

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sympy [A]  time = 4.01, size = 76, normalized size = 1.25 \begin {gather*} - \frac {2 A a^{2}}{5 x^{\frac {5}{2}}} - \frac {4 A a b}{\sqrt {x}} + \frac {2 A b^{2} x^{\frac {3}{2}}}{3} - \frac {2 B a^{2}}{\sqrt {x}} + \frac {4 B a b x^{\frac {3}{2}}}{3} + \frac {2 B b^{2} x^{\frac {7}{2}}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2*(B*x**2+A)/x**(7/2),x)

[Out]

-2*A*a**2/(5*x**(5/2)) - 4*A*a*b/sqrt(x) + 2*A*b**2*x**(3/2)/3 - 2*B*a**2/sqrt(x) + 4*B*a*b*x**(3/2)/3 + 2*B*b
**2*x**(7/2)/7

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