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*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])

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Rubi [A]  time = 0.0535221, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, 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 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]))))

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 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]

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*}

Mathematica [A]  time = 0.0276985, size = 94, normalized size = 0.64 \[ -\frac{2 \left (8 a^2 b^2 x^2 (2 A+7 B x)-2 a^3 b x (4 A+7 B x)+a^4 (5 A+7 B x)+16 a b^3 x^3 (7 B x-4 A)-128 A b^4 x^4\right )}{35 a^5 x^{7/2} \sqrt{a+b x}} \]

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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Maple [A]  time = 0.005, size = 101, normalized size = 0.7 \begin{align*} -{\frac{-256\,A{b}^{4}{x}^{4}+224\,Ba{b}^{3}{x}^{4}-128\,Aa{b}^{3}{x}^{3}+112\,B{a}^{2}{b}^{2}{x}^{3}+32\,A{a}^{2}{b}^{2}{x}^{2}-28\,B{a}^{3}b{x}^{2}-16\,A{a}^{3}bx+14\,B{a}^{4}x+10\,A{a}^{4}}{35\,{a}^{5}}{x}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{bx+a}}}} \end{align*}

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)/x^(7/2)/(b*x+a)^(1/2)/a^5

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 2.62902, size = 251, normalized size = 1.71 \begin{align*} -\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 )}} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [A]  time = 1.35809, size = 304, normalized size = 2.07 \begin{align*} \frac{{\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^{4} b^{12}} - \frac{28 \,{\left (9 \, B a^{11} b^{9}{\left | b \right |} - 11 \, A a^{10} b^{10}{\left | b \right |}\right )}}{a^{4} b^{12}}\right )} + \frac{70 \,{\left (4 \, B a^{12} b^{9}{\left | b \right |} - 5 \, A a^{11} b^{10}{\left | b \right |}\right )}}{a^{4} b^{12}}\right )} - \frac{35 \,{\left (3 \, B a^{13} b^{9}{\left | b \right |} - 4 \, A a^{12} b^{10}{\left | b \right |}\right )}}{a^{4} b^{12}}\right )} \sqrt{b x + a}}{26880 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{7}{2}}} - \frac{4 \,{\left (B a b^{\frac{9}{2}} - A b^{\frac{11}{2}}\right )}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} a^{4}{\left | b \right |}} \end{align*}

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]

1/26880*((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^4*b^12) - 28*(9*B*a^
11*b^9*abs(b) - 11*A*a^10*b^10*abs(b))/(a^4*b^12)) + 70*(4*B*a^12*b^9*abs(b) - 5*A*a^11*b^10*abs(b))/(a^4*b^12
)) - 35*(3*B*a^13*b^9*abs(b) - 4*A*a^12*b^10*abs(b))/(a^4*b^12))*sqrt(b*x + a)/((b*x + a)*b - a*b)^(7/2) - 4*(
B*a*b^(9/2) - A*b^(11/2))/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)*a^4*abs(b))