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

Optimal. Leaf size=147 \[ \frac {32 b^2 \sqrt {a+b x} (8 A b-7 a B)}{35 a^5 \sqrt {x}}-\frac {16 b \sqrt {a+b x} (8 A b-7 a B)}{35 a^4 x^{3/2}}+\frac {12 \sqrt {a+b x} (8 A b-7 a B)}{35 a^3 x^{5/2}}-\frac {2 (8 A b-7 a B)}{7 a^2 x^{5/2} \sqrt {a+b x}}-\frac {2 A}{7 a x^{7/2} \sqrt {a+b x}} \]

[Out]

-2/7*A/a/x^(7/2)/(b*x+a)^(1/2)-2/7*(8*A*b-7*B*a)/a^2/x^(5/2)/(b*x+a)^(1/2)+12/35*(8*A*b-7*B*a)*(b*x+a)^(1/2)/a
^3/x^(5/2)-16/35*b*(8*A*b-7*B*a)*(b*x+a)^(1/2)/a^4/x^(3/2)+32/35*b^2*(8*A*b-7*B*a)*(b*x+a)^(1/2)/a^5/x^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {78, 45, 37} \[ \frac {32 b^2 \sqrt {a+b x} (8 A b-7 a B)}{35 a^5 \sqrt {x}}-\frac {16 b \sqrt {a+b x} (8 A b-7 a B)}{35 a^4 x^{3/2}}+\frac {12 \sqrt {a+b x} (8 A b-7 a B)}{35 a^3 x^{5/2}}-\frac {2 (8 A b-7 a B)}{7 a^2 x^{5/2} \sqrt {a+b x}}-\frac {2 A}{7 a x^{7/2} \sqrt {a+b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A)/(7*a*x^(7/2)*Sqrt[a + b*x]) - (2*(8*A*b - 7*a*B))/(7*a^2*x^(5/2)*Sqrt[a + b*x]) + (12*(8*A*b - 7*a*B)*S
qrt[a + b*x])/(35*a^3*x^(5/2)) - (16*b*(8*A*b - 7*a*B)*Sqrt[a + b*x])/(35*a^4*x^(3/2)) + (32*b^2*(8*A*b - 7*a*
B)*Sqrt[a + b*x])/(35*a^5*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} (a+b x)^{3/2}} \, dx &=-\frac {2 A}{7 a x^{7/2} \sqrt {a+b x}}+\frac {\left (2 \left (-4 A b+\frac {7 a B}{2}\right )\right ) \int \frac {1}{x^{7/2} (a+b x)^{3/2}} \, dx}{7 a}\\ &=-\frac {2 A}{7 a x^{7/2} \sqrt {a+b x}}-\frac {2 (8 A b-7 a B)}{7 a^2 x^{5/2} \sqrt {a+b x}}-\frac {(6 (8 A b-7 a B)) \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx}{7 a^2}\\ &=-\frac {2 A}{7 a x^{7/2} \sqrt {a+b x}}-\frac {2 (8 A b-7 a B)}{7 a^2 x^{5/2} \sqrt {a+b x}}+\frac {12 (8 A b-7 a B) \sqrt {a+b x}}{35 a^3 x^{5/2}}+\frac {(24 b (8 A b-7 a B)) \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{35 a^3}\\ &=-\frac {2 A}{7 a x^{7/2} \sqrt {a+b x}}-\frac {2 (8 A b-7 a B)}{7 a^2 x^{5/2} \sqrt {a+b x}}+\frac {12 (8 A b-7 a B) \sqrt {a+b x}}{35 a^3 x^{5/2}}-\frac {16 b (8 A b-7 a B) \sqrt {a+b x}}{35 a^4 x^{3/2}}-\frac {\left (16 b^2 (8 A b-7 a B)\right ) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{35 a^4}\\ &=-\frac {2 A}{7 a x^{7/2} \sqrt {a+b x}}-\frac {2 (8 A b-7 a B)}{7 a^2 x^{5/2} \sqrt {a+b x}}+\frac {12 (8 A b-7 a B) \sqrt {a+b x}}{35 a^3 x^{5/2}}-\frac {16 b (8 A b-7 a B) \sqrt {a+b x}}{35 a^4 x^{3/2}}+\frac {32 b^2 (8 A b-7 a B) \sqrt {a+b x}}{35 a^5 \sqrt {x}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 94, normalized size = 0.64 \[ -\frac {2 \left (a^4 (5 A+7 B x)-2 a^3 b x (4 A+7 B x)+8 a^2 b^2 x^2 (2 A+7 B x)+16 a b^3 x^3 (7 B x-4 A)-128 A b^4 x^4\right )}{35 a^5 x^{7/2} \sqrt {a+b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(-128*A*b^4*x^4 + 16*a*b^3*x^3*(-4*A + 7*B*x) + 8*a^2*b^2*x^2*(2*A + 7*B*x) - 2*a^3*b*x*(4*A + 7*B*x) + a^
4*(5*A + 7*B*x)))/(35*a^5*x^(7/2)*Sqrt[a + b*x])

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fricas [A]  time = 0.54, size = 116, normalized size = 0.79 \[ -\frac {2 \, {\left (5 \, A a^{4} + 16 \, {\left (7 \, B a b^{3} - 8 \, A b^{4}\right )} x^{4} + 8 \, {\left (7 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{3} - 2 \, {\left (7 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{2} + {\left (7 \, B a^{4} - 8 \, A a^{3} b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{35 \, {\left (a^{5} b x^{5} + a^{6} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.59, size = 292, normalized size = 1.99 \[ -\frac {2 \, {\left ({\left (b x + a\right )} {\left ({\left (b x + a\right )} {\left (\frac {{\left (77 \, B a^{10} b^{9} {\left | b \right |} - 93 \, A a^{9} b^{10} {\left | b \right |}\right )} {\left (b x + a\right )}}{a^{14} b^{4}} - \frac {28 \, {\left (9 \, B a^{11} b^{9} {\left | b \right |} - 11 \, A a^{10} b^{10} {\left | b \right |}\right )}}{a^{14} b^{4}}\right )} + \frac {70 \, {\left (4 \, B a^{12} b^{9} {\left | b \right |} - 5 \, A a^{11} b^{10} {\left | b \right |}\right )}}{a^{14} b^{4}}\right )} - \frac {35 \, {\left (3 \, B a^{13} b^{9} {\left | b \right |} - 4 \, A a^{12} b^{10} {\left | b \right |}\right )}}{a^{14} b^{4}}\right )} \sqrt {b x + a}}{35 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {7}{2}}} - \frac {4 \, {\left (B^{2} a^{2} b^{9} - 2 \, A B a b^{10} + A^{2} b^{11}\right )}}{{\left (B a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {9}{2}} + B a^{2} b^{\frac {11}{2}} - A {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {11}{2}} - A a b^{\frac {13}{2}}\right )} a^{4} {\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/35*((b*x + a)*((b*x + a)*((77*B*a^10*b^9*abs(b) - 93*A*a^9*b^10*abs(b))*(b*x + a)/(a^14*b^4) - 28*(9*B*a^11
*b^9*abs(b) - 11*A*a^10*b^10*abs(b))/(a^14*b^4)) + 70*(4*B*a^12*b^9*abs(b) - 5*A*a^11*b^10*abs(b))/(a^14*b^4))
- 35*(3*B*a^13*b^9*abs(b) - 4*A*a^12*b^10*abs(b))/(a^14*b^4))*sqrt(b*x + a)/((b*x + a)*b - a*b)^(7/2) - 4*(B^
2*a^2*b^9 - 2*A*B*a*b^10 + A^2*b^11)/((B*a*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(9/2) + B*a^2
*b^(11/2) - A*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(11/2) - A*a*b^(13/2))*a^4*abs(b))

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maple [A]  time = 0.01, size = 101, normalized size = 0.69 \[ -\frac {2 \left (-128 A \,b^{4} x^{4}+112 B a \,b^{3} x^{4}-64 A a \,b^{3} x^{3}+56 B \,a^{2} b^{2} x^{3}+16 A \,a^{2} b^{2} x^{2}-14 B \,a^{3} b \,x^{2}-8 A \,a^{3} b x +7 B \,a^{4} x +5 A \,a^{4}\right )}{35 \sqrt {b x +a}\, a^{5} x^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/35*(-128*A*b^4*x^4+112*B*a*b^3*x^4-64*A*a*b^3*x^3+56*B*a^2*b^2*x^3+16*A*a^2*b^2*x^2-14*B*a^3*b*x^2-8*A*a^3*
b*x+7*B*a^4*x+5*A*a^4)/(b*x+a)^(1/2)/x^(7/2)/a^5

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maxima [A]  time = 0.87, size = 188, normalized size = 1.28 \[ -\frac {32 \, B b^{3} x}{5 \, \sqrt {b x^{2} + a x} a^{4}} + \frac {256 \, A b^{4} x}{35 \, \sqrt {b x^{2} + a x} a^{5}} - \frac {16 \, B b^{2}}{5 \, \sqrt {b x^{2} + a x} a^{3}} + \frac {128 \, A b^{3}}{35 \, \sqrt {b x^{2} + a x} a^{4}} + \frac {4 \, B b}{5 \, \sqrt {b x^{2} + a x} a^{2} x} - \frac {32 \, A b^{2}}{35 \, \sqrt {b x^{2} + a x} a^{3} x} - \frac {2 \, B}{5 \, \sqrt {b x^{2} + a x} a x^{2}} + \frac {16 \, A b}{35 \, \sqrt {b x^{2} + a x} a^{2} x^{2}} - \frac {2 \, A}{7 \, \sqrt {b x^{2} + a x} a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-32/5*B*b^3*x/(sqrt(b*x^2 + a*x)*a^4) + 256/35*A*b^4*x/(sqrt(b*x^2 + a*x)*a^5) - 16/5*B*b^2/(sqrt(b*x^2 + a*x)
*a^3) + 128/35*A*b^3/(sqrt(b*x^2 + a*x)*a^4) + 4/5*B*b/(sqrt(b*x^2 + a*x)*a^2*x) - 32/35*A*b^2/(sqrt(b*x^2 + a
*x)*a^3*x) - 2/5*B/(sqrt(b*x^2 + a*x)*a*x^2) + 16/35*A*b/(sqrt(b*x^2 + a*x)*a^2*x^2) - 2/7*A/(sqrt(b*x^2 + a*x
)*a*x^3)

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mupad [B]  time = 0.94, size = 116, normalized size = 0.79 \[ -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{7\,a\,b}+\frac {4\,x^2\,\left (8\,A\,b-7\,B\,a\right )}{35\,a^3}-\frac {x^4\,\left (256\,A\,b^4-224\,B\,a\,b^3\right )}{35\,a^5\,b}-\frac {16\,b\,x^3\,\left (8\,A\,b-7\,B\,a\right )}{35\,a^4}+\frac {x\,\left (14\,B\,a^4-16\,A\,a^3\,b\right )}{35\,a^5\,b}\right )}{x^{9/2}+\frac {a\,x^{7/2}}{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((a + b*x)^(1/2)*((2*A)/(7*a*b) + (4*x^2*(8*A*b - 7*B*a))/(35*a^3) - (x^4*(256*A*b^4 - 224*B*a*b^3))/(35*a^5*
b) - (16*b*x^3*(8*A*b - 7*B*a))/(35*a^4) + (x*(14*B*a^4 - 16*A*a^3*b))/(35*a^5*b)))/(x^(9/2) + (a*x^(7/2))/b)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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