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

Optimal. Leaf size=150 \[ -\frac{32 b^3 (a+b x)^{3/2} (8 A b-11 a B)}{3465 a^5 x^{3/2}}+\frac{16 b^2 (a+b x)^{3/2} (8 A b-11 a B)}{1155 a^4 x^{5/2}}-\frac{4 b (a+b x)^{3/2} (8 A b-11 a B)}{231 a^3 x^{7/2}}+\frac{2 (a+b x)^{3/2} (8 A b-11 a B)}{99 a^2 x^{9/2}}-\frac{2 A (a+b x)^{3/2}}{11 a x^{11/2}} \]

[Out]

(-2*A*(a + b*x)^(3/2))/(11*a*x^(11/2)) + (2*(8*A*b - 11*a*B)*(a + b*x)^(3/2))/(99*a^2*x^(9/2)) - (4*b*(8*A*b -
11*a*B)*(a + b*x)^(3/2))/(231*a^3*x^(7/2)) + (16*b^2*(8*A*b - 11*a*B)*(a + b*x)^(3/2))/(1155*a^4*x^(5/2)) - (
32*b^3*(8*A*b - 11*a*B)*(a + b*x)^(3/2))/(3465*a^5*x^(3/2))

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Rubi [A]  time = 0.0553233, antiderivative size = 150, 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} \[ -\frac{32 b^3 (a+b x)^{3/2} (8 A b-11 a B)}{3465 a^5 x^{3/2}}+\frac{16 b^2 (a+b x)^{3/2} (8 A b-11 a B)}{1155 a^4 x^{5/2}}-\frac{4 b (a+b x)^{3/2} (8 A b-11 a B)}{231 a^3 x^{7/2}}+\frac{2 (a+b x)^{3/2} (8 A b-11 a B)}{99 a^2 x^{9/2}}-\frac{2 A (a+b x)^{3/2}}{11 a x^{11/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A*(a + b*x)^(3/2))/(11*a*x^(11/2)) + (2*(8*A*b - 11*a*B)*(a + b*x)^(3/2))/(99*a^2*x^(9/2)) - (4*b*(8*A*b -
11*a*B)*(a + b*x)^(3/2))/(231*a^3*x^(7/2)) + (16*b^2*(8*A*b - 11*a*B)*(a + b*x)^(3/2))/(1155*a^4*x^(5/2)) - (
32*b^3*(8*A*b - 11*a*B)*(a + b*x)^(3/2))/(3465*a^5*x^(3/2))

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{\sqrt{a+b x} (A+B x)}{x^{13/2}} \, dx &=-\frac{2 A (a+b x)^{3/2}}{11 a x^{11/2}}+\frac{\left (2 \left (-4 A b+\frac{11 a B}{2}\right )\right ) \int \frac{\sqrt{a+b x}}{x^{11/2}} \, dx}{11 a}\\ &=-\frac{2 A (a+b x)^{3/2}}{11 a x^{11/2}}+\frac{2 (8 A b-11 a B) (a+b x)^{3/2}}{99 a^2 x^{9/2}}+\frac{(2 b (8 A b-11 a B)) \int \frac{\sqrt{a+b x}}{x^{9/2}} \, dx}{33 a^2}\\ &=-\frac{2 A (a+b x)^{3/2}}{11 a x^{11/2}}+\frac{2 (8 A b-11 a B) (a+b x)^{3/2}}{99 a^2 x^{9/2}}-\frac{4 b (8 A b-11 a B) (a+b x)^{3/2}}{231 a^3 x^{7/2}}-\frac{\left (8 b^2 (8 A b-11 a B)\right ) \int \frac{\sqrt{a+b x}}{x^{7/2}} \, dx}{231 a^3}\\ &=-\frac{2 A (a+b x)^{3/2}}{11 a x^{11/2}}+\frac{2 (8 A b-11 a B) (a+b x)^{3/2}}{99 a^2 x^{9/2}}-\frac{4 b (8 A b-11 a B) (a+b x)^{3/2}}{231 a^3 x^{7/2}}+\frac{16 b^2 (8 A b-11 a B) (a+b x)^{3/2}}{1155 a^4 x^{5/2}}+\frac{\left (16 b^3 (8 A b-11 a B)\right ) \int \frac{\sqrt{a+b x}}{x^{5/2}} \, dx}{1155 a^4}\\ &=-\frac{2 A (a+b x)^{3/2}}{11 a x^{11/2}}+\frac{2 (8 A b-11 a B) (a+b x)^{3/2}}{99 a^2 x^{9/2}}-\frac{4 b (8 A b-11 a B) (a+b x)^{3/2}}{231 a^3 x^{7/2}}+\frac{16 b^2 (8 A b-11 a B) (a+b x)^{3/2}}{1155 a^4 x^{5/2}}-\frac{32 b^3 (8 A b-11 a B) (a+b x)^{3/2}}{3465 a^5 x^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0324762, size = 95, normalized size = 0.63 \[ -\frac{2 (a+b x)^{3/2} \left (24 a^2 b^2 x^2 (10 A+11 B x)-10 a^3 b x (28 A+33 B x)+35 a^4 (9 A+11 B x)-16 a b^3 x^3 (12 A+11 B x)+128 A b^4 x^4\right )}{3465 a^5 x^{11/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a + b*x)^(3/2)*(128*A*b^4*x^4 + 35*a^4*(9*A + 11*B*x) + 24*a^2*b^2*x^2*(10*A + 11*B*x) - 16*a*b^3*x^3*(12
*A + 11*B*x) - 10*a^3*b*x*(28*A + 33*B*x)))/(3465*a^5*x^(11/2))

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Maple [A]  time = 0.005, size = 101, normalized size = 0.7 \begin{align*} -{\frac{256\,A{b}^{4}{x}^{4}-352\,Ba{b}^{3}{x}^{4}-384\,Aa{b}^{3}{x}^{3}+528\,B{a}^{2}{b}^{2}{x}^{3}+480\,A{a}^{2}{b}^{2}{x}^{2}-660\,B{a}^{3}b{x}^{2}-560\,A{a}^{3}bx+770\,B{a}^{4}x+630\,A{a}^{4}}{3465\,{a}^{5}} \left ( bx+a \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3465*(b*x+a)^(3/2)*(128*A*b^4*x^4-176*B*a*b^3*x^4-192*A*a*b^3*x^3+264*B*a^2*b^2*x^3+240*A*a^2*b^2*x^2-330*B
*a^3*b*x^2-280*A*a^3*b*x+385*B*a^4*x+315*A*a^4)/x^(11/2)/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)*(b*x+a)^(1/2)/x^(13/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.5296, size = 293, normalized size = 1.95 \begin{align*} -\frac{2 \,{\left (315 \, A a^{5} - 16 \,{\left (11 \, B a b^{4} - 8 \, A b^{5}\right )} x^{5} + 8 \,{\left (11 \, B a^{2} b^{3} - 8 \, A a b^{4}\right )} x^{4} - 6 \,{\left (11 \, B a^{3} b^{2} - 8 \, A a^{2} b^{3}\right )} x^{3} + 5 \,{\left (11 \, B a^{4} b - 8 \, A a^{3} b^{2}\right )} x^{2} + 35 \,{\left (11 \, B a^{5} + A a^{4} b\right )} x\right )} \sqrt{b x + a}}{3465 \, a^{5} x^{\frac{11}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3465*(315*A*a^5 - 16*(11*B*a*b^4 - 8*A*b^5)*x^5 + 8*(11*B*a^2*b^3 - 8*A*a*b^4)*x^4 - 6*(11*B*a^3*b^2 - 8*A*
a^2*b^3)*x^3 + 5*(11*B*a^4*b - 8*A*a^3*b^2)*x^2 + 35*(11*B*a^5 + A*a^4*b)*x)*sqrt(b*x + a)/(a^5*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)*(b*x+a)**(1/2)/x**(13/2),x)

[Out]

Timed out

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Giac [A]  time = 1.31333, size = 248, normalized size = 1.65 \begin{align*} -\frac{{\left ({\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (11 \, B a b^{10} - 8 \, A b^{11}\right )}{\left (b x + a\right )}}{a^{6} b^{18}} - \frac{11 \,{\left (11 \, B a^{2} b^{10} - 8 \, A a b^{11}\right )}}{a^{6} b^{18}}\right )} + \frac{99 \,{\left (11 \, B a^{3} b^{10} - 8 \, A a^{2} b^{11}\right )}}{a^{6} b^{18}}\right )} - \frac{231 \,{\left (11 \, B a^{4} b^{10} - 8 \, A a^{3} b^{11}\right )}}{a^{6} b^{18}}\right )}{\left (b x + a\right )} + \frac{1155 \,{\left (B a^{5} b^{10} - A a^{4} b^{11}\right )}}{a^{6} b^{18}}\right )}{\left (b x + a\right )}^{\frac{3}{2}} 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)*(b*x+a)^(1/2)/x^(13/2),x, algorithm="giac")

[Out]

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