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

Optimal. Leaf size=117 \[ \frac {16 b^2 \sqrt {a+b x} (6 A b-7 a B)}{105 a^4 \sqrt {x}}-\frac {8 b \sqrt {a+b x} (6 A b-7 a B)}{105 a^3 x^{3/2}}+\frac {2 \sqrt {a+b x} (6 A b-7 a B)}{35 a^2 x^{5/2}}-\frac {2 A \sqrt {a+b x}}{7 a x^{7/2}} \]

[Out]

-2/7*A*(b*x+a)^(1/2)/a/x^(7/2)+2/35*(6*A*b-7*B*a)*(b*x+a)^(1/2)/a^2/x^(5/2)-8/105*b*(6*A*b-7*B*a)*(b*x+a)^(1/2
)/a^3/x^(3/2)+16/105*b^2*(6*A*b-7*B*a)*(b*x+a)^(1/2)/a^4/x^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {78, 45, 37} \[ \frac {16 b^2 \sqrt {a+b x} (6 A b-7 a B)}{105 a^4 \sqrt {x}}-\frac {8 b \sqrt {a+b x} (6 A b-7 a B)}{105 a^3 x^{3/2}}+\frac {2 \sqrt {a+b x} (6 A b-7 a B)}{35 a^2 x^{5/2}}-\frac {2 A \sqrt {a+b x}}{7 a x^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A*Sqrt[a + b*x])/(7*a*x^(7/2)) + (2*(6*A*b - 7*a*B)*Sqrt[a + b*x])/(35*a^2*x^(5/2)) - (8*b*(6*A*b - 7*a*B)
*Sqrt[a + b*x])/(105*a^3*x^(3/2)) + (16*b^2*(6*A*b - 7*a*B)*Sqrt[a + b*x])/(105*a^4*Sqrt[x])

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 {A+B x}{x^{9/2} \sqrt {a+b x}} \, dx &=-\frac {2 A \sqrt {a+b x}}{7 a x^{7/2}}+\frac {\left (2 \left (-3 A b+\frac {7 a B}{2}\right )\right ) \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx}{7 a}\\ &=-\frac {2 A \sqrt {a+b x}}{7 a x^{7/2}}+\frac {2 (6 A b-7 a B) \sqrt {a+b x}}{35 a^2 x^{5/2}}+\frac {(4 b (6 A b-7 a B)) \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{35 a^2}\\ &=-\frac {2 A \sqrt {a+b x}}{7 a x^{7/2}}+\frac {2 (6 A b-7 a B) \sqrt {a+b x}}{35 a^2 x^{5/2}}-\frac {8 b (6 A b-7 a B) \sqrt {a+b x}}{105 a^3 x^{3/2}}-\frac {\left (8 b^2 (6 A b-7 a B)\right ) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{105 a^3}\\ &=-\frac {2 A \sqrt {a+b x}}{7 a x^{7/2}}+\frac {2 (6 A b-7 a B) \sqrt {a+b x}}{35 a^2 x^{5/2}}-\frac {8 b (6 A b-7 a B) \sqrt {a+b x}}{105 a^3 x^{3/2}}+\frac {16 b^2 (6 A b-7 a B) \sqrt {a+b x}}{105 a^4 \sqrt {x}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 76, normalized size = 0.65 \[ -\frac {2 \sqrt {a+b x} \left (3 a^3 (5 A+7 B x)-2 a^2 b x (9 A+14 B x)+8 a b^2 x^2 (3 A+7 B x)-48 A b^3 x^3\right )}{105 a^4 x^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a + b*x]*(-48*A*b^3*x^3 + 8*a*b^2*x^2*(3*A + 7*B*x) + 3*a^3*(5*A + 7*B*x) - 2*a^2*b*x*(9*A + 14*B*x))
)/(105*a^4*x^(7/2))

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fricas [A]  time = 0.68, size = 78, normalized size = 0.67 \[ -\frac {2 \, {\left (15 \, A a^{3} + 8 \, {\left (7 \, B a b^{2} - 6 \, A b^{3}\right )} x^{3} - 4 \, {\left (7 \, B a^{2} b - 6 \, A a b^{2}\right )} x^{2} + 3 \, {\left (7 \, B a^{3} - 6 \, A a^{2} b\right )} x\right )} \sqrt {b x + a}}{105 \, a^{4} x^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/105*(15*A*a^3 + 8*(7*B*a*b^2 - 6*A*b^3)*x^3 - 4*(7*B*a^2*b - 6*A*a*b^2)*x^2 + 3*(7*B*a^3 - 6*A*a^2*b)*x)*sq
rt(b*x + a)/(a^4*x^(7/2))

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giac [A]  time = 1.57, size = 137, normalized size = 1.17 \[ -\frac {2 \, {\left ({\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (7 \, B a b^{6} - 6 \, A b^{7}\right )} {\left (b x + a\right )}}{a^{4}} - \frac {7 \, {\left (7 \, B a^{2} b^{6} - 6 \, A a b^{7}\right )}}{a^{4}}\right )} + \frac {35 \, {\left (7 \, B a^{3} b^{6} - 6 \, A a^{2} b^{7}\right )}}{a^{4}}\right )} - \frac {105 \, {\left (B a^{4} b^{6} - A a^{3} b^{7}\right )}}{a^{4}}\right )} \sqrt {b x + a} b}{105 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {7}{2}} {\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/105*((b*x + a)*(4*(b*x + a)*(2*(7*B*a*b^6 - 6*A*b^7)*(b*x + a)/a^4 - 7*(7*B*a^2*b^6 - 6*A*a*b^7)/a^4) + 35*
(7*B*a^3*b^6 - 6*A*a^2*b^7)/a^4) - 105*(B*a^4*b^6 - A*a^3*b^7)/a^4)*sqrt(b*x + a)*b/(((b*x + a)*b - a*b)^(7/2)
*abs(b))

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maple [A]  time = 0.01, size = 77, normalized size = 0.66 \[ -\frac {2 \sqrt {b x +a}\, \left (-48 A \,b^{3} x^{3}+56 B a \,b^{2} x^{3}+24 A a \,b^{2} x^{2}-28 B \,a^{2} b \,x^{2}-18 A \,a^{2} b x +21 B \,a^{3} x +15 A \,a^{3}\right )}{105 a^{4} x^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/105*(b*x+a)^(1/2)*(-48*A*b^3*x^3+56*B*a*b^2*x^3+24*A*a*b^2*x^2-28*B*a^2*b*x^2-18*A*a^2*b*x+21*B*a^3*x+15*A*
a^3)/x^(7/2)/a^4

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maxima [A]  time = 0.91, size = 152, normalized size = 1.30 \[ -\frac {16 \, \sqrt {b x^{2} + a x} B b^{2}}{15 \, a^{3} x} + \frac {32 \, \sqrt {b x^{2} + a x} A b^{3}}{35 \, a^{4} x} + \frac {8 \, \sqrt {b x^{2} + a x} B b}{15 \, a^{2} x^{2}} - \frac {16 \, \sqrt {b x^{2} + a x} A b^{2}}{35 \, a^{3} x^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} B}{5 \, a x^{3}} + \frac {12 \, \sqrt {b x^{2} + a x} A b}{35 \, a^{2} x^{3}} - \frac {2 \, \sqrt {b x^{2} + a x} A}{7 \, a x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16/15*sqrt(b*x^2 + a*x)*B*b^2/(a^3*x) + 32/35*sqrt(b*x^2 + a*x)*A*b^3/(a^4*x) + 8/15*sqrt(b*x^2 + a*x)*B*b/(a
^2*x^2) - 16/35*sqrt(b*x^2 + a*x)*A*b^2/(a^3*x^2) - 2/5*sqrt(b*x^2 + a*x)*B/(a*x^3) + 12/35*sqrt(b*x^2 + a*x)*
A*b/(a^2*x^3) - 2/7*sqrt(b*x^2 + a*x)*A/(a*x^4)

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mupad [B]  time = 0.90, size = 79, normalized size = 0.68 \[ -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{7\,a}+\frac {x\,\left (42\,B\,a^3-36\,A\,a^2\,b\right )}{105\,a^4}-\frac {x^3\,\left (96\,A\,b^3-112\,B\,a\,b^2\right )}{105\,a^4}+\frac {8\,b\,x^2\,\left (6\,A\,b-7\,B\,a\right )}{105\,a^3}\right )}{x^{7/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((a + b*x)^(1/2)*((2*A)/(7*a) + (x*(42*B*a^3 - 36*A*a^2*b))/(105*a^4) - (x^3*(96*A*b^3 - 112*B*a*b^2))/(105*a
^4) + (8*b*x^2*(6*A*b - 7*B*a))/(105*a^3)))/x^(7/2)

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sympy [B]  time = 59.76, size = 796, normalized size = 6.80 \[ - \frac {10 A a^{6} b^{\frac {19}{2}} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} - \frac {18 A a^{5} b^{\frac {21}{2}} x \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} - \frac {10 A a^{4} b^{\frac {23}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} + \frac {10 A a^{3} b^{\frac {25}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} + \frac {60 A a^{2} b^{\frac {27}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} + \frac {80 A a b^{\frac {29}{2}} x^{5} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} + \frac {32 A b^{\frac {31}{2}} x^{6} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} - \frac {6 B a^{4} b^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac {4 B a^{3} b^{\frac {11}{2}} x \sqrt {\frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac {6 B a^{2} b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac {24 B a b^{\frac {15}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac {16 B b^{\frac {17}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-10*A*a**6*b**(19/2)*sqrt(a/(b*x) + 1)/(35*a**7*b**9*x**3 + 105*a**6*b**10*x**4 + 105*a**5*b**11*x**5 + 35*a**
4*b**12*x**6) - 18*A*a**5*b**(21/2)*x*sqrt(a/(b*x) + 1)/(35*a**7*b**9*x**3 + 105*a**6*b**10*x**4 + 105*a**5*b*
*11*x**5 + 35*a**4*b**12*x**6) - 10*A*a**4*b**(23/2)*x**2*sqrt(a/(b*x) + 1)/(35*a**7*b**9*x**3 + 105*a**6*b**1
0*x**4 + 105*a**5*b**11*x**5 + 35*a**4*b**12*x**6) + 10*A*a**3*b**(25/2)*x**3*sqrt(a/(b*x) + 1)/(35*a**7*b**9*
x**3 + 105*a**6*b**10*x**4 + 105*a**5*b**11*x**5 + 35*a**4*b**12*x**6) + 60*A*a**2*b**(27/2)*x**4*sqrt(a/(b*x)
+ 1)/(35*a**7*b**9*x**3 + 105*a**6*b**10*x**4 + 105*a**5*b**11*x**5 + 35*a**4*b**12*x**6) + 80*A*a*b**(29/2)*
x**5*sqrt(a/(b*x) + 1)/(35*a**7*b**9*x**3 + 105*a**6*b**10*x**4 + 105*a**5*b**11*x**5 + 35*a**4*b**12*x**6) +
32*A*b**(31/2)*x**6*sqrt(a/(b*x) + 1)/(35*a**7*b**9*x**3 + 105*a**6*b**10*x**4 + 105*a**5*b**11*x**5 + 35*a**4
*b**12*x**6) - 6*B*a**4*b**(9/2)*sqrt(a/(b*x) + 1)/(15*a**5*b**4*x**2 + 30*a**4*b**5*x**3 + 15*a**3*b**6*x**4)
- 4*B*a**3*b**(11/2)*x*sqrt(a/(b*x) + 1)/(15*a**5*b**4*x**2 + 30*a**4*b**5*x**3 + 15*a**3*b**6*x**4) - 6*B*a*
*2*b**(13/2)*x**2*sqrt(a/(b*x) + 1)/(15*a**5*b**4*x**2 + 30*a**4*b**5*x**3 + 15*a**3*b**6*x**4) - 24*B*a*b**(1
5/2)*x**3*sqrt(a/(b*x) + 1)/(15*a**5*b**4*x**2 + 30*a**4*b**5*x**3 + 15*a**3*b**6*x**4) - 16*B*b**(17/2)*x**4*
sqrt(a/(b*x) + 1)/(15*a**5*b**4*x**2 + 30*a**4*b**5*x**3 + 15*a**3*b**6*x**4)

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