3.536 \(\int \frac{A+B x}{x^{13/2} (a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=213 \[ -\frac{256 b^3 \sqrt{a+b x} (12 A b-11 a B)}{693 a^6 x^{3/2}}+\frac{64 b^2 \sqrt{a+b x} (12 A b-11 a B)}{231 a^5 x^{5/2}}+\frac{512 b^4 \sqrt{a+b x} (12 A b-11 a B)}{693 a^7 \sqrt{x}}-\frac{160 b \sqrt{a+b x} (12 A b-11 a B)}{693 a^4 x^{7/2}}+\frac{20 \sqrt{a+b x} (12 A b-11 a B)}{99 a^3 x^{9/2}}-\frac{2 (12 A b-11 a B)}{11 a^2 x^{9/2} \sqrt{a+b x}}-\frac{2 A}{11 a x^{11/2} \sqrt{a+b x}} \]

[Out]

(-2*A)/(11*a*x^(11/2)*Sqrt[a + b*x]) - (2*(12*A*b - 11*a*B))/(11*a^2*x^(9/2)*Sqrt[a + b*x]) + (20*(12*A*b - 11
*a*B)*Sqrt[a + b*x])/(99*a^3*x^(9/2)) - (160*b*(12*A*b - 11*a*B)*Sqrt[a + b*x])/(693*a^4*x^(7/2)) + (64*b^2*(1
2*A*b - 11*a*B)*Sqrt[a + b*x])/(231*a^5*x^(5/2)) - (256*b^3*(12*A*b - 11*a*B)*Sqrt[a + b*x])/(693*a^6*x^(3/2))
+ (512*b^4*(12*A*b - 11*a*B)*Sqrt[a + b*x])/(693*a^7*Sqrt[x])

________________________________________________________________________________________

Rubi [A]  time = 0.0893804, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {78, 45, 37} \[ -\frac{256 b^3 \sqrt{a+b x} (12 A b-11 a B)}{693 a^6 x^{3/2}}+\frac{64 b^2 \sqrt{a+b x} (12 A b-11 a B)}{231 a^5 x^{5/2}}+\frac{512 b^4 \sqrt{a+b x} (12 A b-11 a B)}{693 a^7 \sqrt{x}}-\frac{160 b \sqrt{a+b x} (12 A b-11 a B)}{693 a^4 x^{7/2}}+\frac{20 \sqrt{a+b x} (12 A b-11 a B)}{99 a^3 x^{9/2}}-\frac{2 (12 A b-11 a B)}{11 a^2 x^{9/2} \sqrt{a+b x}}-\frac{2 A}{11 a x^{11/2} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^(13/2)*(a + b*x)^(3/2)),x]

[Out]

(-2*A)/(11*a*x^(11/2)*Sqrt[a + b*x]) - (2*(12*A*b - 11*a*B))/(11*a^2*x^(9/2)*Sqrt[a + b*x]) + (20*(12*A*b - 11
*a*B)*Sqrt[a + b*x])/(99*a^3*x^(9/2)) - (160*b*(12*A*b - 11*a*B)*Sqrt[a + b*x])/(693*a^4*x^(7/2)) + (64*b^2*(1
2*A*b - 11*a*B)*Sqrt[a + b*x])/(231*a^5*x^(5/2)) - (256*b^3*(12*A*b - 11*a*B)*Sqrt[a + b*x])/(693*a^6*x^(3/2))
+ (512*b^4*(12*A*b - 11*a*B)*Sqrt[a + b*x])/(693*a^7*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^{13/2} (a+b x)^{3/2}} \, dx &=-\frac{2 A}{11 a x^{11/2} \sqrt{a+b x}}+\frac{\left (2 \left (-6 A b+\frac{11 a B}{2}\right )\right ) \int \frac{1}{x^{11/2} (a+b x)^{3/2}} \, dx}{11 a}\\ &=-\frac{2 A}{11 a x^{11/2} \sqrt{a+b x}}-\frac{2 (12 A b-11 a B)}{11 a^2 x^{9/2} \sqrt{a+b x}}-\frac{(10 (12 A b-11 a B)) \int \frac{1}{x^{11/2} \sqrt{a+b x}} \, dx}{11 a^2}\\ &=-\frac{2 A}{11 a x^{11/2} \sqrt{a+b x}}-\frac{2 (12 A b-11 a B)}{11 a^2 x^{9/2} \sqrt{a+b x}}+\frac{20 (12 A b-11 a B) \sqrt{a+b x}}{99 a^3 x^{9/2}}+\frac{(80 b (12 A b-11 a B)) \int \frac{1}{x^{9/2} \sqrt{a+b x}} \, dx}{99 a^3}\\ &=-\frac{2 A}{11 a x^{11/2} \sqrt{a+b x}}-\frac{2 (12 A b-11 a B)}{11 a^2 x^{9/2} \sqrt{a+b x}}+\frac{20 (12 A b-11 a B) \sqrt{a+b x}}{99 a^3 x^{9/2}}-\frac{160 b (12 A b-11 a B) \sqrt{a+b x}}{693 a^4 x^{7/2}}-\frac{\left (160 b^2 (12 A b-11 a B)\right ) \int \frac{1}{x^{7/2} \sqrt{a+b x}} \, dx}{231 a^4}\\ &=-\frac{2 A}{11 a x^{11/2} \sqrt{a+b x}}-\frac{2 (12 A b-11 a B)}{11 a^2 x^{9/2} \sqrt{a+b x}}+\frac{20 (12 A b-11 a B) \sqrt{a+b x}}{99 a^3 x^{9/2}}-\frac{160 b (12 A b-11 a B) \sqrt{a+b x}}{693 a^4 x^{7/2}}+\frac{64 b^2 (12 A b-11 a B) \sqrt{a+b x}}{231 a^5 x^{5/2}}+\frac{\left (128 b^3 (12 A b-11 a B)\right ) \int \frac{1}{x^{5/2} \sqrt{a+b x}} \, dx}{231 a^5}\\ &=-\frac{2 A}{11 a x^{11/2} \sqrt{a+b x}}-\frac{2 (12 A b-11 a B)}{11 a^2 x^{9/2} \sqrt{a+b x}}+\frac{20 (12 A b-11 a B) \sqrt{a+b x}}{99 a^3 x^{9/2}}-\frac{160 b (12 A b-11 a B) \sqrt{a+b x}}{693 a^4 x^{7/2}}+\frac{64 b^2 (12 A b-11 a B) \sqrt{a+b x}}{231 a^5 x^{5/2}}-\frac{256 b^3 (12 A b-11 a B) \sqrt{a+b x}}{693 a^6 x^{3/2}}-\frac{\left (256 b^4 (12 A b-11 a B)\right ) \int \frac{1}{x^{3/2} \sqrt{a+b x}} \, dx}{693 a^6}\\ &=-\frac{2 A}{11 a x^{11/2} \sqrt{a+b x}}-\frac{2 (12 A b-11 a B)}{11 a^2 x^{9/2} \sqrt{a+b x}}+\frac{20 (12 A b-11 a B) \sqrt{a+b x}}{99 a^3 x^{9/2}}-\frac{160 b (12 A b-11 a B) \sqrt{a+b x}}{693 a^4 x^{7/2}}+\frac{64 b^2 (12 A b-11 a B) \sqrt{a+b x}}{231 a^5 x^{5/2}}-\frac{256 b^3 (12 A b-11 a B) \sqrt{a+b x}}{693 a^6 x^{3/2}}+\frac{512 b^4 (12 A b-11 a B) \sqrt{a+b x}}{693 a^7 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.0384034, size = 133, normalized size = 0.62 \[ -\frac{2 \left (8 a^4 b^2 x^2 (15 A+22 B x)-32 a^3 b^3 x^3 (6 A+11 B x)+128 a^2 b^4 x^4 (3 A+11 B x)-2 a^5 b x (42 A+55 B x)+7 a^6 (9 A+11 B x)+256 a b^5 x^5 (11 B x-6 A)-3072 A b^6 x^6\right )}{693 a^7 x^{11/2} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^(13/2)*(a + b*x)^(3/2)),x]

[Out]

(-2*(-3072*A*b^6*x^6 + 256*a*b^5*x^5*(-6*A + 11*B*x) + 128*a^2*b^4*x^4*(3*A + 11*B*x) - 32*a^3*b^3*x^3*(6*A +
11*B*x) + 7*a^6*(9*A + 11*B*x) + 8*a^4*b^2*x^2*(15*A + 22*B*x) - 2*a^5*b*x*(42*A + 55*B*x)))/(693*a^7*x^(11/2)
*Sqrt[a + b*x])

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 149, normalized size = 0.7 \begin{align*} -{\frac{-6144\,A{b}^{6}{x}^{6}+5632\,Ba{b}^{5}{x}^{6}-3072\,Aa{b}^{5}{x}^{5}+2816\,B{a}^{2}{b}^{4}{x}^{5}+768\,A{a}^{2}{b}^{4}{x}^{4}-704\,B{a}^{3}{b}^{3}{x}^{4}-384\,A{a}^{3}{b}^{3}{x}^{3}+352\,B{a}^{4}{b}^{2}{x}^{3}+240\,A{a}^{4}{b}^{2}{x}^{2}-220\,B{a}^{5}b{x}^{2}-168\,A{a}^{5}bx+154\,B{a}^{6}x+126\,A{a}^{6}}{693\,{a}^{7}}{x}^{-{\frac{11}{2}}}{\frac{1}{\sqrt{bx+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^(13/2)/(b*x+a)^(3/2),x)

[Out]

-2/693*(-3072*A*b^6*x^6+2816*B*a*b^5*x^6-1536*A*a*b^5*x^5+1408*B*a^2*b^4*x^5+384*A*a^2*b^4*x^4-352*B*a^3*b^3*x
^4-192*A*a^3*b^3*x^3+176*B*a^4*b^2*x^3+120*A*a^4*b^2*x^2-110*B*a^5*b*x^2-84*A*a^5*b*x+77*B*a^6*x+63*A*a^6)/x^(
11/2)/(b*x+a)^(1/2)/a^7

________________________________________________________________________________________

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^(13/2)/(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.69164, size = 378, normalized size = 1.77 \begin{align*} -\frac{2 \,{\left (63 \, A a^{6} + 256 \,{\left (11 \, B a b^{5} - 12 \, A b^{6}\right )} x^{6} + 128 \,{\left (11 \, B a^{2} b^{4} - 12 \, A a b^{5}\right )} x^{5} - 32 \,{\left (11 \, B a^{3} b^{3} - 12 \, A a^{2} b^{4}\right )} x^{4} + 16 \,{\left (11 \, B a^{4} b^{2} - 12 \, A a^{3} b^{3}\right )} x^{3} - 10 \,{\left (11 \, B a^{5} b - 12 \, A a^{4} b^{2}\right )} x^{2} + 7 \,{\left (11 \, B a^{6} - 12 \, A a^{5} b\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{693 \,{\left (a^{7} b x^{7} + a^{8} x^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(13/2)/(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

-2/693*(63*A*a^6 + 256*(11*B*a*b^5 - 12*A*b^6)*x^6 + 128*(11*B*a^2*b^4 - 12*A*a*b^5)*x^5 - 32*(11*B*a^3*b^3 -
12*A*a^2*b^4)*x^4 + 16*(11*B*a^4*b^2 - 12*A*a^3*b^3)*x^3 - 10*(11*B*a^5*b - 12*A*a^4*b^2)*x^2 + 7*(11*B*a^6 -
12*A*a^5*b)*x)*sqrt(b*x + a)*sqrt(x)/(a^7*b*x^7 + a^8*x^6)

________________________________________________________________________________________

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**(13/2)/(b*x+a)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.49876, size = 406, normalized size = 1.91 \begin{align*} \frac{{\left ({\left ({\left ({\left (b x + a\right )}{\left ({\left (b x + a\right )}{\left (\frac{{\left (2123 \, B a^{21} b^{15}{\left | b \right |} - 2379 \, A a^{20} b^{16}{\left | b \right |}\right )}{\left (b x + a\right )}}{a^{6} b^{18}} - \frac{22 \,{\left (515 \, B a^{22} b^{15}{\left | b \right |} - 579 \, A a^{21} b^{16}{\left | b \right |}\right )}}{a^{6} b^{18}}\right )} + \frac{99 \,{\left (247 \, B a^{23} b^{15}{\left | b \right |} - 279 \, A a^{22} b^{16}{\left | b \right |}\right )}}{a^{6} b^{18}}\right )} - \frac{924 \,{\left (29 \, B a^{24} b^{15}{\left | b \right |} - 33 \, A a^{23} b^{16}{\left | b \right |}\right )}}{a^{6} b^{18}}\right )}{\left (b x + a\right )} + \frac{1155 \,{\left (13 \, B a^{25} b^{15}{\left | b \right |} - 15 \, A a^{24} b^{16}{\left | b \right |}\right )}}{a^{6} b^{18}}\right )}{\left (b x + a\right )} - \frac{693 \,{\left (5 \, B a^{26} b^{15}{\left | b \right |} - 6 \, A a^{25} b^{16}{\left | b \right |}\right )}}{a^{6} b^{18}}\right )} \sqrt{b x + a}}{2838528 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{11}{2}}} - \frac{4 \,{\left (B a b^{\frac{13}{2}} - A b^{\frac{15}{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^{6}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(13/2)/(b*x+a)^(3/2),x, algorithm="giac")

[Out]

1/2838528*((((b*x + a)*((b*x + a)*((2123*B*a^21*b^15*abs(b) - 2379*A*a^20*b^16*abs(b))*(b*x + a)/(a^6*b^18) -
22*(515*B*a^22*b^15*abs(b) - 579*A*a^21*b^16*abs(b))/(a^6*b^18)) + 99*(247*B*a^23*b^15*abs(b) - 279*A*a^22*b^1
6*abs(b))/(a^6*b^18)) - 924*(29*B*a^24*b^15*abs(b) - 33*A*a^23*b^16*abs(b))/(a^6*b^18))*(b*x + a) + 1155*(13*B
*a^25*b^15*abs(b) - 15*A*a^24*b^16*abs(b))/(a^6*b^18))*(b*x + a) - 693*(5*B*a^26*b^15*abs(b) - 6*A*a^25*b^16*a
bs(b))/(a^6*b^18))*sqrt(b*x + a)/((b*x + a)*b - a*b)^(11/2) - 4*(B*a*b^(13/2) - A*b^(15/2))/(((sqrt(b*x + a)*s
qrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)*a^6*abs(b))