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

Optimal. Leaf size=183 \[ -\frac{128 b^3 \sqrt{a+b x} (10 A b-11 a B)}{3465 a^5 x^{3/2}}+\frac{32 b^2 \sqrt{a+b x} (10 A b-11 a B)}{1155 a^4 x^{5/2}}+\frac{256 b^4 \sqrt{a+b x} (10 A b-11 a B)}{3465 a^6 \sqrt{x}}-\frac{16 b \sqrt{a+b x} (10 A b-11 a B)}{693 a^3 x^{7/2}}+\frac{2 \sqrt{a+b x} (10 A b-11 a B)}{99 a^2 x^{9/2}}-\frac{2 A \sqrt{a+b x}}{11 a x^{11/2}} \]

[Out]

(-2*A*Sqrt[a + b*x])/(11*a*x^(11/2)) + (2*(10*A*b - 11*a*B)*Sqrt[a + b*x])/(99*a^2*x^(9/2)) - (16*b*(10*A*b -
11*a*B)*Sqrt[a + b*x])/(693*a^3*x^(7/2)) + (32*b^2*(10*A*b - 11*a*B)*Sqrt[a + b*x])/(1155*a^4*x^(5/2)) - (128*
b^3*(10*A*b - 11*a*B)*Sqrt[a + b*x])/(3465*a^5*x^(3/2)) + (256*b^4*(10*A*b - 11*a*B)*Sqrt[a + b*x])/(3465*a^6*
Sqrt[x])

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Rubi [A]  time = 0.0711368, antiderivative size = 183, 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} \[ -\frac{128 b^3 \sqrt{a+b x} (10 A b-11 a B)}{3465 a^5 x^{3/2}}+\frac{32 b^2 \sqrt{a+b x} (10 A b-11 a B)}{1155 a^4 x^{5/2}}+\frac{256 b^4 \sqrt{a+b x} (10 A b-11 a B)}{3465 a^6 \sqrt{x}}-\frac{16 b \sqrt{a+b x} (10 A b-11 a B)}{693 a^3 x^{7/2}}+\frac{2 \sqrt{a+b x} (10 A b-11 a B)}{99 a^2 x^{9/2}}-\frac{2 A \sqrt{a+b x}}{11 a x^{11/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A*Sqrt[a + b*x])/(11*a*x^(11/2)) + (2*(10*A*b - 11*a*B)*Sqrt[a + b*x])/(99*a^2*x^(9/2)) - (16*b*(10*A*b -
11*a*B)*Sqrt[a + b*x])/(693*a^3*x^(7/2)) + (32*b^2*(10*A*b - 11*a*B)*Sqrt[a + b*x])/(1155*a^4*x^(5/2)) - (128*
b^3*(10*A*b - 11*a*B)*Sqrt[a + b*x])/(3465*a^5*x^(3/2)) + (256*b^4*(10*A*b - 11*a*B)*Sqrt[a + b*x])/(3465*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^{13/2} \sqrt{a+b x}} \, dx &=-\frac{2 A \sqrt{a+b x}}{11 a x^{11/2}}+\frac{\left (2 \left (-5 A b+\frac{11 a B}{2}\right )\right ) \int \frac{1}{x^{11/2} \sqrt{a+b x}} \, dx}{11 a}\\ &=-\frac{2 A \sqrt{a+b x}}{11 a x^{11/2}}+\frac{2 (10 A b-11 a B) \sqrt{a+b x}}{99 a^2 x^{9/2}}+\frac{(8 b (10 A b-11 a B)) \int \frac{1}{x^{9/2} \sqrt{a+b x}} \, dx}{99 a^2}\\ &=-\frac{2 A \sqrt{a+b x}}{11 a x^{11/2}}+\frac{2 (10 A b-11 a B) \sqrt{a+b x}}{99 a^2 x^{9/2}}-\frac{16 b (10 A b-11 a B) \sqrt{a+b x}}{693 a^3 x^{7/2}}-\frac{\left (16 b^2 (10 A b-11 a B)\right ) \int \frac{1}{x^{7/2} \sqrt{a+b x}} \, dx}{231 a^3}\\ &=-\frac{2 A \sqrt{a+b x}}{11 a x^{11/2}}+\frac{2 (10 A b-11 a B) \sqrt{a+b x}}{99 a^2 x^{9/2}}-\frac{16 b (10 A b-11 a B) \sqrt{a+b x}}{693 a^3 x^{7/2}}+\frac{32 b^2 (10 A b-11 a B) \sqrt{a+b x}}{1155 a^4 x^{5/2}}+\frac{\left (64 b^3 (10 A b-11 a B)\right ) \int \frac{1}{x^{5/2} \sqrt{a+b x}} \, dx}{1155 a^4}\\ &=-\frac{2 A \sqrt{a+b x}}{11 a x^{11/2}}+\frac{2 (10 A b-11 a B) \sqrt{a+b x}}{99 a^2 x^{9/2}}-\frac{16 b (10 A b-11 a B) \sqrt{a+b x}}{693 a^3 x^{7/2}}+\frac{32 b^2 (10 A b-11 a B) \sqrt{a+b x}}{1155 a^4 x^{5/2}}-\frac{128 b^3 (10 A b-11 a B) \sqrt{a+b x}}{3465 a^5 x^{3/2}}-\frac{\left (128 b^4 (10 A b-11 a B)\right ) \int \frac{1}{x^{3/2} \sqrt{a+b x}} \, dx}{3465 a^5}\\ &=-\frac{2 A \sqrt{a+b x}}{11 a x^{11/2}}+\frac{2 (10 A b-11 a B) \sqrt{a+b x}}{99 a^2 x^{9/2}}-\frac{16 b (10 A b-11 a B) \sqrt{a+b x}}{693 a^3 x^{7/2}}+\frac{32 b^2 (10 A b-11 a B) \sqrt{a+b x}}{1155 a^4 x^{5/2}}-\frac{128 b^3 (10 A b-11 a B) \sqrt{a+b x}}{3465 a^5 x^{3/2}}+\frac{256 b^4 (10 A b-11 a B) \sqrt{a+b x}}{3465 a^6 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.0347568, size = 114, normalized size = 0.62 \[ -\frac{2 \sqrt{a+b x} \left (16 a^3 b^2 x^2 (25 A+33 B x)-32 a^2 b^3 x^3 (15 A+22 B x)-10 a^4 b x (35 A+44 B x)+35 a^5 (9 A+11 B x)+128 a b^4 x^4 (5 A+11 B x)-1280 A b^5 x^5\right )}{3465 a^6 x^{11/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a + b*x]*(-1280*A*b^5*x^5 + 128*a*b^4*x^4*(5*A + 11*B*x) + 35*a^5*(9*A + 11*B*x) - 32*a^2*b^3*x^3*(15
*A + 22*B*x) + 16*a^3*b^2*x^2*(25*A + 33*B*x) - 10*a^4*b*x*(35*A + 44*B*x)))/(3465*a^6*x^(11/2))

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Maple [A]  time = 0.006, size = 125, normalized size = 0.7 \begin{align*} -{\frac{-2560\,A{b}^{5}{x}^{5}+2816\,B{x}^{5}a{b}^{4}+1280\,aA{b}^{4}{x}^{4}-1408\,B{x}^{4}{a}^{2}{b}^{3}-960\,{a}^{2}A{b}^{3}{x}^{3}+1056\,B{x}^{3}{a}^{3}{b}^{2}+800\,{a}^{3}A{b}^{2}{x}^{2}-880\,B{x}^{2}{a}^{4}b-700\,{a}^{4}Abx+770\,{a}^{5}Bx+630\,A{a}^{5}}{3465\,{a}^{6}}\sqrt{bx+a}{x}^{-{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^(13/2)/(b*x+a)^(1/2),x)

[Out]

-2/3465*(b*x+a)^(1/2)*(-1280*A*b^5*x^5+1408*B*a*b^4*x^5+640*A*a*b^4*x^4-704*B*a^2*b^3*x^4-480*A*a^2*b^3*x^3+52
8*B*a^3*b^2*x^3+400*A*a^3*b^2*x^2-440*B*a^4*b*x^2-350*A*a^4*b*x+385*B*a^5*x+315*A*a^5)/x^(11/2)/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^(13/2)/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.51206, size = 308, normalized size = 1.68 \begin{align*} -\frac{2 \,{\left (315 \, A a^{5} + 128 \,{\left (11 \, B a b^{4} - 10 \, A b^{5}\right )} x^{5} - 64 \,{\left (11 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} x^{4} + 48 \,{\left (11 \, B a^{3} b^{2} - 10 \, A a^{2} b^{3}\right )} x^{3} - 40 \,{\left (11 \, B a^{4} b - 10 \, A a^{3} b^{2}\right )} x^{2} + 35 \,{\left (11 \, B a^{5} - 10 \, A a^{4} b\right )} x\right )} \sqrt{b x + a}}{3465 \, a^{6} x^{\frac{11}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(13/2)/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

-2/3465*(315*A*a^5 + 128*(11*B*a*b^4 - 10*A*b^5)*x^5 - 64*(11*B*a^2*b^3 - 10*A*a*b^4)*x^4 + 48*(11*B*a^3*b^2 -
10*A*a^2*b^3)*x^3 - 40*(11*B*a^4*b - 10*A*a^3*b^2)*x^2 + 35*(11*B*a^5 - 10*A*a^4*b)*x)*sqrt(b*x + a)/(a^6*x^(
11/2))

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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**(13/2)/(b*x+a)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.22825, size = 296, normalized size = 1.62 \begin{align*} \frac{{\left ({\left (8 \,{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (11 \, B a b^{10} - 10 \, A b^{11}\right )}{\left (b x + a\right )}}{a^{6} b^{18}} - \frac{11 \,{\left (11 \, B a^{2} b^{10} - 10 \, A a b^{11}\right )}}{a^{6} b^{18}}\right )} + \frac{99 \,{\left (11 \, B a^{3} b^{10} - 10 \, A a^{2} b^{11}\right )}}{a^{6} b^{18}}\right )} - \frac{231 \,{\left (11 \, B a^{4} b^{10} - 10 \, A a^{3} b^{11}\right )}}{a^{6} b^{18}}\right )}{\left (b x + a\right )} + \frac{1155 \,{\left (11 \, B a^{5} b^{10} - 10 \, A a^{4} b^{11}\right )}}{a^{6} b^{18}}\right )}{\left (b x + a\right )} - \frac{3465 \,{\left (B a^{6} b^{10} - A a^{5} b^{11}\right )}}{a^{6} b^{18}}\right )} \sqrt{b x + a} b}{14192640 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{11}{2}}{\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)^(1/2),x, algorithm="giac")

[Out]

1/14192640*((8*(2*(b*x + a)*(4*(b*x + a)*(2*(11*B*a*b^10 - 10*A*b^11)*(b*x + a)/(a^6*b^18) - 11*(11*B*a^2*b^10
- 10*A*a*b^11)/(a^6*b^18)) + 99*(11*B*a^3*b^10 - 10*A*a^2*b^11)/(a^6*b^18)) - 231*(11*B*a^4*b^10 - 10*A*a^3*b
^11)/(a^6*b^18))*(b*x + a) + 1155*(11*B*a^5*b^10 - 10*A*a^4*b^11)/(a^6*b^18))*(b*x + a) - 3465*(B*a^6*b^10 - A
*a^5*b^11)/(a^6*b^18))*sqrt(b*x + a)*b/(((b*x + a)*b - a*b)^(11/2)*abs(b))