3.5.72 \(\int \frac {(a+b x)^{3/2} (A+B x)}{x^{9/2}} \, dx\)

Optimal. Leaf size=53 \[ \frac {2 (a+b x)^{5/2} (2 A b-7 a B)}{35 a^2 x^{5/2}}-\frac {2 A (a+b x)^{5/2}}{7 a x^{7/2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {78, 37} \begin {gather*} \frac {2 (a+b x)^{5/2} (2 A b-7 a B)}{35 a^2 x^{5/2}}-\frac {2 A (a+b x)^{5/2}}{7 a x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(3/2)*(A + B*x))/x^(9/2),x]

[Out]

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rubi steps

\begin {align*} \int \frac {(a+b x)^{3/2} (A+B x)}{x^{9/2}} \, dx &=-\frac {2 A (a+b x)^{5/2}}{7 a x^{7/2}}+\frac {\left (2 \left (-A b+\frac {7 a B}{2}\right )\right ) \int \frac {(a+b x)^{3/2}}{x^{7/2}} \, dx}{7 a}\\ &=-\frac {2 A (a+b x)^{5/2}}{7 a x^{7/2}}+\frac {2 (2 A b-7 a B) (a+b x)^{5/2}}{35 a^2 x^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 36, normalized size = 0.68 \begin {gather*} -\frac {2 (a+b x)^{5/2} (5 a A+7 a B x-2 A b x)}{35 a^2 x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(3/2)*(A + B*x))/x^(9/2),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.26, size = 81, normalized size = 1.53 \begin {gather*} -\frac {2 \sqrt {a+b x} \left (5 a^3 A+7 a^3 B x+8 a^2 A b x+14 a^2 b B x^2+a A b^2 x^2+7 a b^2 B x^3-2 A b^3 x^3\right )}{35 a^2 x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b*x)^(3/2)*(A + B*x))/x^(9/2),x]

[Out]

(-2*Sqrt[a + b*x]*(5*a^3*A + 8*a^2*A*b*x + 7*a^3*B*x + a*A*b^2*x^2 + 14*a^2*b*B*x^2 - 2*A*b^3*x^3 + 7*a*b^2*B*
x^3))/(35*a^2*x^(7/2))

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fricas [A]  time = 1.67, size = 74, normalized size = 1.40 \begin {gather*} -\frac {2 \, {\left (5 \, A a^{3} + {\left (7 \, B a b^{2} - 2 \, A b^{3}\right )} x^{3} + {\left (14 \, B a^{2} b + A a b^{2}\right )} x^{2} + {\left (7 \, B a^{3} + 8 \, A a^{2} b\right )} x\right )} \sqrt {b x + a}}{35 \, a^{2} x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(B*x+A)/x^(9/2),x, algorithm="fricas")

[Out]

-2/35*(5*A*a^3 + (7*B*a*b^2 - 2*A*b^3)*x^3 + (14*B*a^2*b + A*a*b^2)*x^2 + (7*B*a^3 + 8*A*a^2*b)*x)*sqrt(b*x +
a)/(a^2*x^(7/2))

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giac [A]  time = 1.95, size = 78, normalized size = 1.47 \begin {gather*} -\frac {2 \, {\left (b x + a\right )}^{\frac {5}{2}} b {\left (\frac {{\left (7 \, B a^{2} b^{6} - 2 \, A a b^{7}\right )} {\left (b x + a\right )}}{a^{3}} - \frac {7 \, {\left (B a^{3} b^{6} - A a^{2} b^{7}\right )}}{a^{3}}\right )}}{35 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {7}{2}} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(B*x+A)/x^(9/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 31, normalized size = 0.58 \begin {gather*} -\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (-2 A x b +7 B a x +5 A a \right )}{35 a^{2} x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(3/2)*(B*x+A)/x^(9/2),x)

[Out]

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

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maxima [B]  time = 0.89, size = 176, normalized size = 3.32 \begin {gather*} -\frac {2 \, \sqrt {b x^{2} + a x} B b^{2}}{5 \, a x} + \frac {4 \, \sqrt {b x^{2} + a x} A b^{3}}{35 \, a^{2} x} + \frac {\sqrt {b x^{2} + a x} B b}{5 \, x^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} A b^{2}}{35 \, a x^{2}} + \frac {3 \, \sqrt {b x^{2} + a x} B a}{5 \, x^{3}} + \frac {3 \, \sqrt {b x^{2} + a x} A b}{70 \, x^{3}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B}{x^{4}} + \frac {3 \, \sqrt {b x^{2} + a x} A a}{14 \, x^{4}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A}{2 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(B*x+A)/x^(9/2),x, algorithm="maxima")

[Out]

-2/5*sqrt(b*x^2 + a*x)*B*b^2/(a*x) + 4/35*sqrt(b*x^2 + a*x)*A*b^3/(a^2*x) + 1/5*sqrt(b*x^2 + a*x)*B*b/x^2 - 2/
35*sqrt(b*x^2 + a*x)*A*b^2/(a*x^2) + 3/5*sqrt(b*x^2 + a*x)*B*a/x^3 + 3/70*sqrt(b*x^2 + a*x)*A*b/x^3 - (b*x^2 +
a*x)^(3/2)*B/x^4 + 3/14*sqrt(b*x^2 + a*x)*A*a/x^4 - 1/2*(b*x^2 + a*x)^(3/2)*A/x^5

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mupad [B]  time = 0.76, size = 76, normalized size = 1.43 \begin {gather*} -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A\,a}{7}+\frac {x\,\left (14\,B\,a^3+16\,A\,b\,a^2\right )}{35\,a^2}-\frac {x^3\,\left (4\,A\,b^3-14\,B\,a\,b^2\right )}{35\,a^2}+\frac {2\,b\,x^2\,\left (A\,b+14\,B\,a\right )}{35\,a}\right )}{x^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^(3/2))/x^(9/2),x)

[Out]

-((a + b*x)^(1/2)*((2*A*a)/7 + (x*(14*B*a^3 + 16*A*a^2*b))/(35*a^2) - (x^3*(4*A*b^3 - 14*B*a*b^2))/(35*a^2) +
(2*b*x^2*(A*b + 14*B*a))/(35*a)))/x^(7/2)

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sympy [B]  time = 144.28, size = 158, normalized size = 2.98 \begin {gather*} A \left (- \frac {2 a \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{7 x^{3}} - \frac {16 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{35 x^{2}} - \frac {2 b^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}}{35 a x} + \frac {4 b^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}}{35 a^{2}}\right ) + B \left (- \frac {2 a \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{5 x^{2}} - \frac {4 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{5 x} - \frac {2 b^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}}{5 a}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(3/2)*(B*x+A)/x**(9/2),x)

[Out]

A*(-2*a*sqrt(b)*sqrt(a/(b*x) + 1)/(7*x**3) - 16*b**(3/2)*sqrt(a/(b*x) + 1)/(35*x**2) - 2*b**(5/2)*sqrt(a/(b*x)
+ 1)/(35*a*x) + 4*b**(7/2)*sqrt(a/(b*x) + 1)/(35*a**2)) + B*(-2*a*sqrt(b)*sqrt(a/(b*x) + 1)/(5*x**2) - 4*b**(
3/2)*sqrt(a/(b*x) + 1)/(5*x) - 2*b**(5/2)*sqrt(a/(b*x) + 1)/(5*a))

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