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

Optimal. Leaf size=61 \[ -\frac {2 a^2 A}{\sqrt {x}}+\frac {2}{11} b x^{11/2} (2 a B+A b)+\frac {2}{5} a x^{5/2} (a B+2 A b)+\frac {2}{17} b^2 B x^{17/2} \]

[Out]

2/5*a*(2*A*b+B*a)*x^(5/2)+2/11*b*(A*b+2*B*a)*x^(11/2)+2/17*b^2*B*x^(17/2)-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} \[ -\frac {2 a^2 A}{\sqrt {x}}+\frac {2}{11} b x^{11/2} (2 a B+A b)+\frac {2}{5} a x^{5/2} (a B+2 A b)+\frac {2}{17} b^2 B x^{17/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^2*A)/Sqrt[x] + (2*a*(2*A*b + a*B)*x^(5/2))/5 + (2*b*(A*b + 2*a*B)*x^(11/2))/11 + (2*b^2*B*x^(17/2))/17

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^{3/2}} \, dx &=\int \left (\frac {a^2 A}{x^{3/2}}+a (2 A b+a B) x^{3/2}+b (A b+2 a B) x^{9/2}+b^2 B x^{15/2}\right ) \, dx\\ &=-\frac {2 a^2 A}{\sqrt {x}}+\frac {2}{5} a (2 A b+a B) x^{5/2}+\frac {2}{11} b (A b+2 a B) x^{11/2}+\frac {2}{17} b^2 B x^{17/2}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 60, normalized size = 0.98 \[ \frac {-374 a^2 \left (5 A-B x^3\right )+68 a b x^3 \left (11 A+5 B x^3\right )+10 b^2 x^6 \left (17 A+11 B x^3\right )}{935 \sqrt {x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-374*a^2*(5*A - B*x^3) + 68*a*b*x^3*(11*A + 5*B*x^3) + 10*b^2*x^6*(17*A + 11*B*x^3))/(935*Sqrt[x])

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fricas [A]  time = 0.57, size = 53, normalized size = 0.87 \[ \frac {2 \, {\left (55 \, B b^{2} x^{9} + 85 \, {\left (2 \, B a b + A b^{2}\right )} x^{6} + 187 \, {\left (B a^{2} + 2 \, A a b\right )} x^{3} - 935 \, A a^{2}\right )}}{935 \, \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/935*(55*B*b^2*x^9 + 85*(2*B*a*b + A*b^2)*x^6 + 187*(B*a^2 + 2*A*a*b)*x^3 - 935*A*a^2)/sqrt(x)

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giac [A]  time = 0.15, size = 53, normalized size = 0.87 \[ \frac {2}{17} \, B b^{2} x^{\frac {17}{2}} + \frac {4}{11} \, B a b x^{\frac {11}{2}} + \frac {2}{11} \, A b^{2} x^{\frac {11}{2}} + \frac {2}{5} \, B a^{2} x^{\frac {5}{2}} + \frac {4}{5} \, A a b x^{\frac {5}{2}} - \frac {2 \, A a^{2}}{\sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/17*B*b^2*x^(17/2) + 4/11*B*a*b*x^(11/2) + 2/11*A*b^2*x^(11/2) + 2/5*B*a^2*x^(5/2) + 4/5*A*a*b*x^(5/2) - 2*A*
a^2/sqrt(x)

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maple [A]  time = 0.05, size = 56, normalized size = 0.92 \[ -\frac {2 \left (-55 b^{2} B \,x^{9}-85 A \,b^{2} x^{6}-170 B a b \,x^{6}-374 A a b \,x^{3}-187 B \,a^{2} x^{3}+935 a^{2} A \right )}{935 \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/935*(-55*B*b^2*x^9-85*A*b^2*x^6-170*B*a*b*x^6-374*A*a*b*x^3-187*B*a^2*x^3+935*A*a^2)/x^(1/2)

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maxima [A]  time = 0.54, size = 51, normalized size = 0.84 \[ \frac {2}{17} \, B b^{2} x^{\frac {17}{2}} + \frac {2}{11} \, {\left (2 \, B a b + A b^{2}\right )} x^{\frac {11}{2}} + \frac {2}{5} \, {\left (B a^{2} + 2 \, A a b\right )} x^{\frac {5}{2}} - \frac {2 \, A a^{2}}{\sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/17*B*b^2*x^(17/2) + 2/11*(2*B*a*b + A*b^2)*x^(11/2) + 2/5*(B*a^2 + 2*A*a*b)*x^(5/2) - 2*A*a^2/sqrt(x)

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mupad [B]  time = 0.05, size = 51, normalized size = 0.84 \[ x^{5/2}\,\left (\frac {2\,B\,a^2}{5}+\frac {4\,A\,b\,a}{5}\right )+x^{11/2}\,\left (\frac {2\,A\,b^2}{11}+\frac {4\,B\,a\,b}{11}\right )-\frac {2\,A\,a^2}{\sqrt {x}}+\frac {2\,B\,b^2\,x^{17/2}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(5/2)*((2*B*a^2)/5 + (4*A*a*b)/5) + x^(11/2)*((2*A*b^2)/11 + (4*B*a*b)/11) - (2*A*a^2)/x^(1/2) + (2*B*b^2*x^
(17/2))/17

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sympy [A]  time = 7.80, size = 78, normalized size = 1.28 \[ - \frac {2 A a^{2}}{\sqrt {x}} + \frac {4 A a b x^{\frac {5}{2}}}{5} + \frac {2 A b^{2} x^{\frac {11}{2}}}{11} + \frac {2 B a^{2} x^{\frac {5}{2}}}{5} + \frac {4 B a b x^{\frac {11}{2}}}{11} + \frac {2 B b^{2} x^{\frac {17}{2}}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*A*a**2/sqrt(x) + 4*A*a*b*x**(5/2)/5 + 2*A*b**2*x**(11/2)/11 + 2*B*a**2*x**(5/2)/5 + 4*B*a*b*x**(11/2)/11 +
2*B*b**2*x**(17/2)/17

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