3.146 \(\int \frac {(a+b x^3)^2 (A+B x^3)}{x^{7/2}} \, dx\)

Optimal. Leaf size=61 \[ -\frac {2 a^2 A}{5 x^{5/2}}+\frac {2}{7} b x^{7/2} (2 a B+A b)+2 a \sqrt {x} (a B+2 A b)+\frac {2}{13} b^2 B x^{13/2} \]

[Out]

-2/5*a^2*A/x^(5/2)+2/7*b*(A*b+2*B*a)*x^(7/2)+2/13*b^2*B*x^(13/2)+2*a*(2*A*b+B*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} \[ -\frac {2 a^2 A}{5 x^{5/2}}+\frac {2}{7} b x^{7/2} (2 a B+A b)+2 a \sqrt {x} (a B+2 A b)+\frac {2}{13} b^2 B x^{13/2} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^2*(A + B*x^3))/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^(7/2))/7 + (2*b^2*B*x^(13/2))/13

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

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Mathematica [A]  time = 0.02, size = 57, normalized size = 0.93 \[ \frac {2 \left (-91 a^2 \left (A-5 B x^3\right )+130 a b x^3 \left (7 A+B x^3\right )+5 b^2 x^6 \left (13 A+7 B x^3\right )\right )}{455 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^2*(A + B*x^3))/x^(7/2),x]

[Out]

(2*(-91*a^2*(A - 5*B*x^3) + 130*a*b*x^3*(7*A + B*x^3) + 5*b^2*x^6*(13*A + 7*B*x^3)))/(455*x^(5/2))

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fricas [A]  time = 0.82, size = 53, normalized size = 0.87 \[ \frac {2 \, {\left (35 \, B b^{2} x^{9} + 65 \, {\left (2 \, B a b + A b^{2}\right )} x^{6} + 455 \, {\left (B a^{2} + 2 \, A a b\right )} x^{3} - 91 \, A a^{2}\right )}}{455 \, x^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*(B*x^3+A)/x^(7/2),x, algorithm="fricas")

[Out]

2/455*(35*B*b^2*x^9 + 65*(2*B*a*b + A*b^2)*x^6 + 455*(B*a^2 + 2*A*a*b)*x^3 - 91*A*a^2)/x^(5/2)

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giac [A]  time = 0.15, size = 53, normalized size = 0.87 \[ \frac {2}{13} \, B b^{2} x^{\frac {13}{2}} + \frac {4}{7} \, B a b x^{\frac {7}{2}} + \frac {2}{7} \, A b^{2} x^{\frac {7}{2}} + 2 \, B a^{2} \sqrt {x} + 4 \, A a b \sqrt {x} - \frac {2 \, A a^{2}}{5 \, x^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*(B*x^3+A)/x^(7/2),x, algorithm="giac")

[Out]

2/13*B*b^2*x^(13/2) + 4/7*B*a*b*x^(7/2) + 2/7*A*b^2*x^(7/2) + 2*B*a^2*sqrt(x) + 4*A*a*b*sqrt(x) - 2/5*A*a^2/x^
(5/2)

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maple [A]  time = 0.05, size = 56, normalized size = 0.92 \[ -\frac {2 \left (-35 b^{2} B \,x^{9}-65 A \,b^{2} x^{6}-130 B a b \,x^{6}-910 A a b \,x^{3}-455 B \,a^{2} x^{3}+91 a^{2} A \right )}{455 x^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2*(B*x^3+A)/x^(7/2),x)

[Out]

-2/455*(-35*B*b^2*x^9-65*A*b^2*x^6-130*B*a*b*x^6-910*A*a*b*x^3-455*B*a^2*x^3+91*A*a^2)/x^(5/2)

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maxima [A]  time = 0.46, size = 51, normalized size = 0.84 \[ \frac {2}{13} \, B b^{2} x^{\frac {13}{2}} + \frac {2}{7} \, {\left (2 \, B a b + A b^{2}\right )} x^{\frac {7}{2}} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} \sqrt {x} - \frac {2 \, A a^{2}}{5 \, x^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*(B*x^3+A)/x^(7/2),x, algorithm="maxima")

[Out]

2/13*B*b^2*x^(13/2) + 2/7*(2*B*a*b + A*b^2)*x^(7/2) + 2*(B*a^2 + 2*A*a*b)*sqrt(x) - 2/5*A*a^2/x^(5/2)

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mupad [B]  time = 0.05, size = 51, normalized size = 0.84 \[ \sqrt {x}\,\left (2\,B\,a^2+4\,A\,b\,a\right )+x^{7/2}\,\left (\frac {2\,A\,b^2}{7}+\frac {4\,B\,a\,b}{7}\right )-\frac {2\,A\,a^2}{5\,x^{5/2}}+\frac {2\,B\,b^2\,x^{13/2}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^3)*(a + b*x^3)^2)/x^(7/2),x)

[Out]

x^(1/2)*(2*B*a^2 + 4*A*a*b) + x^(7/2)*((2*A*b^2)/7 + (4*B*a*b)/7) - (2*A*a^2)/(5*x^(5/2)) + (2*B*b^2*x^(13/2))
/13

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sympy [A]  time = 18.41, size = 76, normalized size = 1.25 \[ - \frac {2 A a^{2}}{5 x^{\frac {5}{2}}} + 4 A a b \sqrt {x} + \frac {2 A b^{2} x^{\frac {7}{2}}}{7} + 2 B a^{2} \sqrt {x} + \frac {4 B a b x^{\frac {7}{2}}}{7} + \frac {2 B b^{2} x^{\frac {13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**2*(B*x**3+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**(7/2)/7 + 2*B*a**2*sqrt(x) + 4*B*a*b*x**(7/2)/7 + 2*B*b
**2*x**(13/2)/13

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