3.545 \(\int \frac{A+B x}{x^{9/2} (a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=177 \[ \frac{256 b^2 \sqrt{a+b x} (10 A b-7 a B)}{105 a^6 \sqrt{x}}-\frac{128 b \sqrt{a+b x} (10 A b-7 a B)}{105 a^5 x^{3/2}}+\frac{32 \sqrt{a+b x} (10 A b-7 a B)}{35 a^4 x^{5/2}}-\frac{16 (10 A b-7 a B)}{21 a^3 x^{5/2} \sqrt{a+b x}}-\frac{2 (10 A b-7 a B)}{21 a^2 x^{5/2} (a+b x)^{3/2}}-\frac{2 A}{7 a x^{7/2} (a+b x)^{3/2}} \]

[Out]

(-2*A)/(7*a*x^(7/2)*(a + b*x)^(3/2)) - (2*(10*A*b - 7*a*B))/(21*a^2*x^(5/2)*(a + b*x)^(3/2)) - (16*(10*A*b - 7
*a*B))/(21*a^3*x^(5/2)*Sqrt[a + b*x]) + (32*(10*A*b - 7*a*B)*Sqrt[a + b*x])/(35*a^4*x^(5/2)) - (128*b*(10*A*b
- 7*a*B)*Sqrt[a + b*x])/(105*a^5*x^(3/2)) + (256*b^2*(10*A*b - 7*a*B)*Sqrt[a + b*x])/(105*a^6*Sqrt[x])

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Rubi [A]  time = 0.0686382, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {78, 45, 37} \[ \frac{256 b^2 \sqrt{a+b x} (10 A b-7 a B)}{105 a^6 \sqrt{x}}-\frac{128 b \sqrt{a+b x} (10 A b-7 a B)}{105 a^5 x^{3/2}}+\frac{32 \sqrt{a+b x} (10 A b-7 a B)}{35 a^4 x^{5/2}}-\frac{16 (10 A b-7 a B)}{21 a^3 x^{5/2} \sqrt{a+b x}}-\frac{2 (10 A b-7 a B)}{21 a^2 x^{5/2} (a+b x)^{3/2}}-\frac{2 A}{7 a x^{7/2} (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^(9/2)*(a + b*x)^(5/2)),x]

[Out]

(-2*A)/(7*a*x^(7/2)*(a + b*x)^(3/2)) - (2*(10*A*b - 7*a*B))/(21*a^2*x^(5/2)*(a + b*x)^(3/2)) - (16*(10*A*b - 7
*a*B))/(21*a^3*x^(5/2)*Sqrt[a + b*x]) + (32*(10*A*b - 7*a*B)*Sqrt[a + b*x])/(35*a^4*x^(5/2)) - (128*b*(10*A*b
- 7*a*B)*Sqrt[a + b*x])/(105*a^5*x^(3/2)) + (256*b^2*(10*A*b - 7*a*B)*Sqrt[a + b*x])/(105*a^6*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)^{5/2}} \, dx &=-\frac{2 A}{7 a x^{7/2} (a+b x)^{3/2}}+\frac{\left (2 \left (-5 A b+\frac{7 a B}{2}\right )\right ) \int \frac{1}{x^{7/2} (a+b x)^{5/2}} \, dx}{7 a}\\ &=-\frac{2 A}{7 a x^{7/2} (a+b x)^{3/2}}-\frac{2 (10 A b-7 a B)}{21 a^2 x^{5/2} (a+b x)^{3/2}}-\frac{(8 (10 A b-7 a B)) \int \frac{1}{x^{7/2} (a+b x)^{3/2}} \, dx}{21 a^2}\\ &=-\frac{2 A}{7 a x^{7/2} (a+b x)^{3/2}}-\frac{2 (10 A b-7 a B)}{21 a^2 x^{5/2} (a+b x)^{3/2}}-\frac{16 (10 A b-7 a B)}{21 a^3 x^{5/2} \sqrt{a+b x}}-\frac{(16 (10 A b-7 a B)) \int \frac{1}{x^{7/2} \sqrt{a+b x}} \, dx}{7 a^3}\\ &=-\frac{2 A}{7 a x^{7/2} (a+b x)^{3/2}}-\frac{2 (10 A b-7 a B)}{21 a^2 x^{5/2} (a+b x)^{3/2}}-\frac{16 (10 A b-7 a B)}{21 a^3 x^{5/2} \sqrt{a+b x}}+\frac{32 (10 A b-7 a B) \sqrt{a+b x}}{35 a^4 x^{5/2}}+\frac{(64 b (10 A b-7 a B)) \int \frac{1}{x^{5/2} \sqrt{a+b x}} \, dx}{35 a^4}\\ &=-\frac{2 A}{7 a x^{7/2} (a+b x)^{3/2}}-\frac{2 (10 A b-7 a B)}{21 a^2 x^{5/2} (a+b x)^{3/2}}-\frac{16 (10 A b-7 a B)}{21 a^3 x^{5/2} \sqrt{a+b x}}+\frac{32 (10 A b-7 a B) \sqrt{a+b x}}{35 a^4 x^{5/2}}-\frac{128 b (10 A b-7 a B) \sqrt{a+b x}}{105 a^5 x^{3/2}}-\frac{\left (128 b^2 (10 A b-7 a B)\right ) \int \frac{1}{x^{3/2} \sqrt{a+b x}} \, dx}{105 a^5}\\ &=-\frac{2 A}{7 a x^{7/2} (a+b x)^{3/2}}-\frac{2 (10 A b-7 a B)}{21 a^2 x^{5/2} (a+b x)^{3/2}}-\frac{16 (10 A b-7 a B)}{21 a^3 x^{5/2} \sqrt{a+b x}}+\frac{32 (10 A b-7 a B) \sqrt{a+b x}}{35 a^4 x^{5/2}}-\frac{128 b (10 A b-7 a B) \sqrt{a+b x}}{105 a^5 x^{3/2}}+\frac{256 b^2 (10 A b-7 a B) \sqrt{a+b x}}{105 a^6 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.0370019, size = 114, normalized size = 0.64 \[ -\frac{2 \left (16 a^3 b^2 x^2 (5 A+21 B x)+96 a^2 b^3 x^3 (14 B x-5 A)-2 a^4 b x (15 A+28 B x)+3 a^5 (5 A+7 B x)+128 a b^4 x^4 (7 B x-15 A)-1280 A b^5 x^5\right )}{105 a^6 x^{7/2} (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^(9/2)*(a + b*x)^(5/2)),x]

[Out]

(-2*(-1280*A*b^5*x^5 + 128*a*b^4*x^4*(-15*A + 7*B*x) + 3*a^5*(5*A + 7*B*x) + 96*a^2*b^3*x^3*(-5*A + 14*B*x) +
16*a^3*b^2*x^2*(5*A + 21*B*x) - 2*a^4*b*x*(15*A + 28*B*x)))/(105*a^6*x^(7/2)*(a + b*x)^(3/2))

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Maple [A]  time = 0.005, size = 125, normalized size = 0.7 \begin{align*} -{\frac{-2560\,A{b}^{5}{x}^{5}+1792\,B{x}^{5}a{b}^{4}-3840\,aA{b}^{4}{x}^{4}+2688\,B{x}^{4}{a}^{2}{b}^{3}-960\,{a}^{2}A{b}^{3}{x}^{3}+672\,B{x}^{3}{a}^{3}{b}^{2}+160\,{a}^{3}A{b}^{2}{x}^{2}-112\,B{x}^{2}{a}^{4}b-60\,{a}^{4}Abx+42\,{a}^{5}Bx+30\,A{a}^{5}}{105\,{a}^{6}}{x}^{-{\frac{7}{2}}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^(9/2)/(b*x+a)^(5/2),x)

[Out]

-2/105*(-1280*A*b^5*x^5+896*B*a*b^4*x^5-1920*A*a*b^4*x^4+1344*B*a^2*b^3*x^4-480*A*a^2*b^3*x^3+336*B*a^3*b^2*x^
3+80*A*a^3*b^2*x^2-56*B*a^4*b*x^2-30*A*a^4*b*x+21*B*a^5*x+15*A*a^5)/x^(7/2)/(b*x+a)^(3/2)/a^6

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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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.66696, size = 339, normalized size = 1.92 \begin{align*} -\frac{2 \,{\left (15 \, A a^{5} + 128 \,{\left (7 \, B a b^{4} - 10 \, A b^{5}\right )} x^{5} + 192 \,{\left (7 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} x^{4} + 48 \,{\left (7 \, B a^{3} b^{2} - 10 \, A a^{2} b^{3}\right )} x^{3} - 8 \,{\left (7 \, B a^{4} b - 10 \, A a^{3} b^{2}\right )} x^{2} + 3 \,{\left (7 \, B a^{5} - 10 \, A a^{4} b\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{105 \,{\left (a^{6} b^{2} x^{6} + 2 \, a^{7} b x^{5} + a^{8} 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)^(5/2),x, algorithm="fricas")

[Out]

-2/105*(15*A*a^5 + 128*(7*B*a*b^4 - 10*A*b^5)*x^5 + 192*(7*B*a^2*b^3 - 10*A*a*b^4)*x^4 + 48*(7*B*a^3*b^2 - 10*
A*a^2*b^3)*x^3 - 8*(7*B*a^4*b - 10*A*a^3*b^2)*x^2 + 3*(7*B*a^5 - 10*A*a^4*b)*x)*sqrt(b*x + a)*sqrt(x)/(a^6*b^2
*x^6 + 2*a^7*b*x^5 + a^8*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)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.59064, size = 513, normalized size = 2.9 \begin{align*} \frac{{\left ({\left (b x + a\right )}{\left ({\left (b x + a\right )}{\left (\frac{{\left (511 \, B a^{13} b^{9}{\left | b \right |} - 790 \, A a^{12} b^{10}{\left | b \right |}\right )}{\left (b x + a\right )}}{a^{4} b^{12}} - \frac{7 \,{\left (233 \, B a^{14} b^{9}{\left | b \right |} - 365 \, A a^{13} b^{10}{\left | b \right |}\right )}}{a^{4} b^{12}}\right )} + \frac{350 \,{\left (5 \, B a^{15} b^{9}{\left | b \right |} - 8 \, A a^{14} b^{10}{\left | b \right |}\right )}}{a^{4} b^{12}}\right )} - \frac{210 \,{\left (3 \, B a^{16} b^{9}{\left | b \right |} - 5 \, A a^{15} b^{10}{\left | b \right |}\right )}}{a^{4} b^{12}}\right )} \sqrt{b x + a}}{80640 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{7}{2}}} - \frac{4 \,{\left (9 \, B a{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac{9}{2}} + 24 \, B a^{2}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{11}{2}} - 12 \, A{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac{11}{2}} + 11 \, B a^{3} b^{\frac{13}{2}} - 30 \, A a{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{13}{2}} - 14 \, A a^{2} b^{\frac{15}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} a^{5}{\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)^(5/2),x, algorithm="giac")

[Out]

1/80640*((b*x + a)*((b*x + a)*((511*B*a^13*b^9*abs(b) - 790*A*a^12*b^10*abs(b))*(b*x + a)/(a^4*b^12) - 7*(233*
B*a^14*b^9*abs(b) - 365*A*a^13*b^10*abs(b))/(a^4*b^12)) + 350*(5*B*a^15*b^9*abs(b) - 8*A*a^14*b^10*abs(b))/(a^
4*b^12)) - 210*(3*B*a^16*b^9*abs(b) - 5*A*a^15*b^10*abs(b))/(a^4*b^12))*sqrt(b*x + a)/((b*x + a)*b - a*b)^(7/2
) - 4/3*(9*B*a*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*b^(9/2) + 24*B*a^2*(sqrt(b*x + a)*sqrt(b) -
sqrt((b*x + a)*b - a*b))^2*b^(11/2) - 12*A*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*b^(11/2) + 11*
B*a^3*b^(13/2) - 30*A*a*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(13/2) - 14*A*a^2*b^(15/2))/(((s
qrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)^3*a^5*abs(b))