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

Optimal. Leaf size=37 \[ 2 \sqrt {x} (a B+A b)-\frac {2 a A}{5 x^{5/2}}+\frac {2}{7} b B x^{7/2} \]

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Rubi [A]  time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {448} \begin {gather*} 2 \sqrt {x} (a B+A b)-\frac {2 a A}{5 x^{5/2}}+\frac {2}{7} b B x^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.01, size = 36, normalized size = 0.97 \begin {gather*} \frac {2 \left (5 b x^3 \left (7 A+B x^3\right )-7 a \left (A-5 B x^3\right )\right )}{35 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-7*a*(A - 5*B*x^3) + 5*b*x^3*(7*A + B*x^3)))/(35*x^(5/2))

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IntegrateAlgebraic [A]  time = 0.03, size = 35, normalized size = 0.95 \begin {gather*} \frac {2 \left (-7 a A+35 a B x^3+35 A b x^3+5 b B x^6\right )}{35 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-7*a*A + 35*A*b*x^3 + 35*a*B*x^3 + 5*b*B*x^6))/(35*x^(5/2))

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fricas [A]  time = 0.75, size = 29, normalized size = 0.78 \begin {gather*} \frac {2 \, {\left (5 \, B b x^{6} + 35 \, {\left (B a + A b\right )} x^{3} - 7 \, A a\right )}}{35 \, x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(5*B*b*x^6 + 35*(B*a + A*b)*x^3 - 7*A*a)/x^(5/2)

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giac [A]  time = 0.26, size = 29, normalized size = 0.78 \begin {gather*} \frac {2}{7} \, B b x^{\frac {7}{2}} + 2 \, B a \sqrt {x} + 2 \, A b \sqrt {x} - \frac {2 \, A a}{5 \, x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 32, normalized size = 0.86 \begin {gather*} -\frac {2 \left (-5 B b \,x^{6}-35 A b \,x^{3}-35 B a \,x^{3}+7 A a \right )}{35 x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/35*(-5*B*b*x^6-35*A*b*x^3-35*B*a*x^3+7*A*a)/x^(5/2)

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maxima [A]  time = 0.45, size = 27, normalized size = 0.73 \begin {gather*} \frac {2}{7} \, B b x^{\frac {7}{2}} + 2 \, {\left (B a + A b\right )} \sqrt {x} - \frac {2 \, A a}{5 \, x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 30, normalized size = 0.81 \begin {gather*} \frac {2\,A\,b\,x^3-\frac {2\,A\,a}{5}+2\,B\,a\,x^3+\frac {2\,B\,b\,x^6}{7}}{x^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 4.21, size = 42, normalized size = 1.14 \begin {gather*} - \frac {2 A a}{5 x^{\frac {5}{2}}} + 2 A b \sqrt {x} + 2 B a \sqrt {x} + \frac {2 B b x^{\frac {7}{2}}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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