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

Optimal. Leaf size=84 \[ -\frac {4 b (a+b x)^{3/2} (4 A b-7 a B)}{105 a^3 x^{3/2}}+\frac {2 (a+b x)^{3/2} (4 A b-7 a B)}{35 a^2 x^{5/2}}-\frac {2 A (a+b x)^{3/2}}{7 a x^{7/2}} \]

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Rubi [A]  time = 0.03, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {78, 45, 37} \begin {gather*} -\frac {4 b (a+b x)^{3/2} (4 A b-7 a B)}{105 a^3 x^{3/2}}+\frac {2 (a+b x)^{3/2} (4 A b-7 a B)}{35 a^2 x^{5/2}}-\frac {2 A (a+b x)^{3/2}}{7 a x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A*(a + b*x)^(3/2))/(7*a*x^(7/2)) + (2*(4*A*b - 7*a*B)*(a + b*x)^(3/2))/(35*a^2*x^(5/2)) - (4*b*(4*A*b - 7*
a*B)*(a + b*x)^(3/2))/(105*a^3*x^(3/2))

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 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 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 {\sqrt {a+b x} (A+B x)}{x^{9/2}} \, dx &=-\frac {2 A (a+b x)^{3/2}}{7 a x^{7/2}}+\frac {\left (2 \left (-2 A b+\frac {7 a B}{2}\right )\right ) \int \frac {\sqrt {a+b x}}{x^{7/2}} \, dx}{7 a}\\ &=-\frac {2 A (a+b x)^{3/2}}{7 a x^{7/2}}+\frac {2 (4 A b-7 a B) (a+b x)^{3/2}}{35 a^2 x^{5/2}}+\frac {(2 b (4 A b-7 a B)) \int \frac {\sqrt {a+b x}}{x^{5/2}} \, dx}{35 a^2}\\ &=-\frac {2 A (a+b x)^{3/2}}{7 a x^{7/2}}+\frac {2 (4 A b-7 a B) (a+b x)^{3/2}}{35 a^2 x^{5/2}}-\frac {4 b (4 A b-7 a B) (a+b x)^{3/2}}{105 a^3 x^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 57, normalized size = 0.68 \begin {gather*} -\frac {2 (a+b x)^{3/2} \left (3 a^2 (5 A+7 B x)-2 a b x (6 A+7 B x)+8 A b^2 x^2\right )}{105 a^3 x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a + b*x)^(3/2)*(8*A*b^2*x^2 + 3*a^2*(5*A + 7*B*x) - 2*a*b*x*(6*A + 7*B*x)))/(105*a^3*x^(7/2))

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IntegrateAlgebraic [A]  time = 0.18, size = 82, normalized size = 0.98 \begin {gather*} -\frac {2 \sqrt {a+b x} \left (15 a^3 A+21 a^3 B x+3 a^2 A b x+7 a^2 b B x^2-4 a A b^2 x^2-14 a b^2 B x^3+8 A b^3 x^3\right )}{105 a^3 x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a + b*x]*(15*a^3*A + 3*a^2*A*b*x + 21*a^3*B*x - 4*a*A*b^2*x^2 + 7*a^2*b*B*x^2 + 8*A*b^3*x^3 - 14*a*b^
2*B*x^3))/(105*a^3*x^(7/2))

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fricas [A]  time = 1.32, size = 76, normalized size = 0.90 \begin {gather*} -\frac {2 \, {\left (15 \, A a^{3} - 2 \, {\left (7 \, B a b^{2} - 4 \, A b^{3}\right )} x^{3} + {\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{2} + 3 \, {\left (7 \, B a^{3} + A a^{2} b\right )} x\right )} \sqrt {b x + a}}{105 \, a^{3} x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b*x+a)^(1/2)/x^(9/2),x, algorithm="fricas")

[Out]

-2/105*(15*A*a^3 - 2*(7*B*a*b^2 - 4*A*b^3)*x^3 + (7*B*a^2*b - 4*A*a*b^2)*x^2 + 3*(7*B*a^3 + A*a^2*b)*x)*sqrt(b
*x + a)/(a^3*x^(7/2))

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giac [A]  time = 1.33, size = 105, normalized size = 1.25 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}} {\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (7 \, B a b^{6} - 4 \, A b^{7}\right )} {\left (b x + a\right )}}{a^{3}} - \frac {7 \, {\left (7 \, B a^{2} b^{6} - 4 \, A a b^{7}\right )}}{a^{3}}\right )} + \frac {35 \, {\left (B a^{3} b^{6} - A a^{2} b^{7}\right )}}{a^{3}}\right )} b}{105 \, {\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)*(b*x+a)^(1/2)/x^(9/2),x, algorithm="giac")

[Out]

2/105*(b*x + a)^(3/2)*((b*x + a)*(2*(7*B*a*b^6 - 4*A*b^7)*(b*x + a)/a^3 - 7*(7*B*a^2*b^6 - 4*A*a*b^7)/a^3) + 3
5*(B*a^3*b^6 - A*a^2*b^7)/a^3)*b/(((b*x + a)*b - a*b)^(7/2)*abs(b))

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maple [A]  time = 0.00, size = 53, normalized size = 0.63 \begin {gather*} -\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (8 A \,b^{2} x^{2}-14 B a b \,x^{2}-12 A a b x +21 B \,a^{2} x +15 A \,a^{2}\right )}{105 a^{3} x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b*x+a)^(1/2)/x^(9/2),x)

[Out]

-2/105*(b*x+a)^(3/2)*(8*A*b^2*x^2-14*B*a*b*x^2-12*A*a*b*x+21*B*a^2*x+15*A*a^2)/x^(7/2)/a^3

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maxima [B]  time = 0.90, size = 146, normalized size = 1.74 \begin {gather*} \frac {4 \, \sqrt {b x^{2} + a x} B b^{2}}{15 \, a^{2} x} - \frac {16 \, \sqrt {b x^{2} + a x} A b^{3}}{105 \, a^{3} x} - \frac {2 \, \sqrt {b x^{2} + a x} B b}{15 \, a x^{2}} + \frac {8 \, \sqrt {b x^{2} + a x} A b^{2}}{105 \, a^{2} x^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} B}{5 \, x^{3}} - \frac {2 \, \sqrt {b x^{2} + a x} A b}{35 \, a x^{3}} - \frac {2 \, \sqrt {b x^{2} + a x} A}{7 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b*x+a)^(1/2)/x^(9/2),x, algorithm="maxima")

[Out]

4/15*sqrt(b*x^2 + a*x)*B*b^2/(a^2*x) - 16/105*sqrt(b*x^2 + a*x)*A*b^3/(a^3*x) - 2/15*sqrt(b*x^2 + a*x)*B*b/(a*
x^2) + 8/105*sqrt(b*x^2 + a*x)*A*b^2/(a^2*x^2) - 2/5*sqrt(b*x^2 + a*x)*B/x^3 - 2/35*sqrt(b*x^2 + a*x)*A*b/(a*x
^3) - 2/7*sqrt(b*x^2 + a*x)*A/x^4

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mupad [B]  time = 0.70, size = 76, normalized size = 0.90 \begin {gather*} -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{7}+\frac {x\,\left (42\,B\,a^3+6\,A\,b\,a^2\right )}{105\,a^3}+\frac {x^3\,\left (16\,A\,b^3-28\,B\,a\,b^2\right )}{105\,a^3}-\frac {2\,b\,x^2\,\left (4\,A\,b-7\,B\,a\right )}{105\,a^2}\right )}{x^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^(1/2))/x^(9/2),x)

[Out]

-((a + b*x)^(1/2)*((2*A)/7 + (x*(42*B*a^3 + 6*A*a^2*b))/(105*a^3) + (x^3*(16*A*b^3 - 28*B*a*b^2))/(105*a^3) -
(2*b*x^2*(4*A*b - 7*B*a))/(105*a^2)))/x^(7/2)

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sympy [B]  time = 157.54, size = 416, normalized size = 4.95 \begin {gather*} A \left (- \frac {30 a^{5} b^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {66 a^{4} b^{\frac {11}{2}} x \sqrt {\frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {34 a^{3} b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {6 a^{2} b^{\frac {15}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {24 a b^{\frac {17}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {16 b^{\frac {19}{2}} x^{5} \sqrt {\frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}}\right ) + B \left (- \frac {2 \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{5 x^{2}} - \frac {2 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{15 a x} + \frac {4 b^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}}{15 a^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b*x+a)**(1/2)/x**(9/2),x)

[Out]

A*(-30*a**5*b**(9/2)*sqrt(a/(b*x) + 1)/(105*a**5*b**4*x**3 + 210*a**4*b**5*x**4 + 105*a**3*b**6*x**5) - 66*a**
4*b**(11/2)*x*sqrt(a/(b*x) + 1)/(105*a**5*b**4*x**3 + 210*a**4*b**5*x**4 + 105*a**3*b**6*x**5) - 34*a**3*b**(1
3/2)*x**2*sqrt(a/(b*x) + 1)/(105*a**5*b**4*x**3 + 210*a**4*b**5*x**4 + 105*a**3*b**6*x**5) - 6*a**2*b**(15/2)*
x**3*sqrt(a/(b*x) + 1)/(105*a**5*b**4*x**3 + 210*a**4*b**5*x**4 + 105*a**3*b**6*x**5) - 24*a*b**(17/2)*x**4*sq
rt(a/(b*x) + 1)/(105*a**5*b**4*x**3 + 210*a**4*b**5*x**4 + 105*a**3*b**6*x**5) - 16*b**(19/2)*x**5*sqrt(a/(b*x
) + 1)/(105*a**5*b**4*x**3 + 210*a**4*b**5*x**4 + 105*a**3*b**6*x**5)) + B*(-2*sqrt(b)*sqrt(a/(b*x) + 1)/(5*x*
*2) - 2*b**(3/2)*sqrt(a/(b*x) + 1)/(15*a*x) + 4*b**(5/2)*sqrt(a/(b*x) + 1)/(15*a**2))

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