3.489 \(\int \frac{\sqrt{a+b x} (A+B x)}{x^{15/2}} \, dx\)

Optimal. Leaf size=183 \[ \frac{256 b^4 (a+b x)^{3/2} (10 A b-13 a B)}{45045 a^6 x^{3/2}}-\frac{128 b^3 (a+b x)^{3/2} (10 A b-13 a B)}{15015 a^5 x^{5/2}}+\frac{32 b^2 (a+b x)^{3/2} (10 A b-13 a B)}{3003 a^4 x^{7/2}}-\frac{16 b (a+b x)^{3/2} (10 A b-13 a B)}{1287 a^3 x^{9/2}}+\frac{2 (a+b x)^{3/2} (10 A b-13 a B)}{143 a^2 x^{11/2}}-\frac{2 A (a+b x)^{3/2}}{13 a x^{13/2}} \]

[Out]

(-2*A*(a + b*x)^(3/2))/(13*a*x^(13/2)) + (2*(10*A*b - 13*a*B)*(a + b*x)^(3/2))/(143*a^2*x^(11/2)) - (16*b*(10*
A*b - 13*a*B)*(a + b*x)^(3/2))/(1287*a^3*x^(9/2)) + (32*b^2*(10*A*b - 13*a*B)*(a + b*x)^(3/2))/(3003*a^4*x^(7/
2)) - (128*b^3*(10*A*b - 13*a*B)*(a + b*x)^(3/2))/(15015*a^5*x^(5/2)) + (256*b^4*(10*A*b - 13*a*B)*(a + b*x)^(
3/2))/(45045*a^6*x^(3/2))

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Rubi [A]  time = 0.0734041, 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} \[ \frac{256 b^4 (a+b x)^{3/2} (10 A b-13 a B)}{45045 a^6 x^{3/2}}-\frac{128 b^3 (a+b x)^{3/2} (10 A b-13 a B)}{15015 a^5 x^{5/2}}+\frac{32 b^2 (a+b x)^{3/2} (10 A b-13 a B)}{3003 a^4 x^{7/2}}-\frac{16 b (a+b x)^{3/2} (10 A b-13 a B)}{1287 a^3 x^{9/2}}+\frac{2 (a+b x)^{3/2} (10 A b-13 a B)}{143 a^2 x^{11/2}}-\frac{2 A (a+b x)^{3/2}}{13 a x^{13/2}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x]*(A + B*x))/x^(15/2),x]

[Out]

(-2*A*(a + b*x)^(3/2))/(13*a*x^(13/2)) + (2*(10*A*b - 13*a*B)*(a + b*x)^(3/2))/(143*a^2*x^(11/2)) - (16*b*(10*
A*b - 13*a*B)*(a + b*x)^(3/2))/(1287*a^3*x^(9/2)) + (32*b^2*(10*A*b - 13*a*B)*(a + b*x)^(3/2))/(3003*a^4*x^(7/
2)) - (128*b^3*(10*A*b - 13*a*B)*(a + b*x)^(3/2))/(15015*a^5*x^(5/2)) + (256*b^4*(10*A*b - 13*a*B)*(a + b*x)^(
3/2))/(45045*a^6*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^{15/2}} \, dx &=-\frac{2 A (a+b x)^{3/2}}{13 a x^{13/2}}+\frac{\left (2 \left (-5 A b+\frac{13 a B}{2}\right )\right ) \int \frac{\sqrt{a+b x}}{x^{13/2}} \, dx}{13 a}\\ &=-\frac{2 A (a+b x)^{3/2}}{13 a x^{13/2}}+\frac{2 (10 A b-13 a B) (a+b x)^{3/2}}{143 a^2 x^{11/2}}+\frac{(8 b (10 A b-13 a B)) \int \frac{\sqrt{a+b x}}{x^{11/2}} \, dx}{143 a^2}\\ &=-\frac{2 A (a+b x)^{3/2}}{13 a x^{13/2}}+\frac{2 (10 A b-13 a B) (a+b x)^{3/2}}{143 a^2 x^{11/2}}-\frac{16 b (10 A b-13 a B) (a+b x)^{3/2}}{1287 a^3 x^{9/2}}-\frac{\left (16 b^2 (10 A b-13 a B)\right ) \int \frac{\sqrt{a+b x}}{x^{9/2}} \, dx}{429 a^3}\\ &=-\frac{2 A (a+b x)^{3/2}}{13 a x^{13/2}}+\frac{2 (10 A b-13 a B) (a+b x)^{3/2}}{143 a^2 x^{11/2}}-\frac{16 b (10 A b-13 a B) (a+b x)^{3/2}}{1287 a^3 x^{9/2}}+\frac{32 b^2 (10 A b-13 a B) (a+b x)^{3/2}}{3003 a^4 x^{7/2}}+\frac{\left (64 b^3 (10 A b-13 a B)\right ) \int \frac{\sqrt{a+b x}}{x^{7/2}} \, dx}{3003 a^4}\\ &=-\frac{2 A (a+b x)^{3/2}}{13 a x^{13/2}}+\frac{2 (10 A b-13 a B) (a+b x)^{3/2}}{143 a^2 x^{11/2}}-\frac{16 b (10 A b-13 a B) (a+b x)^{3/2}}{1287 a^3 x^{9/2}}+\frac{32 b^2 (10 A b-13 a B) (a+b x)^{3/2}}{3003 a^4 x^{7/2}}-\frac{128 b^3 (10 A b-13 a B) (a+b x)^{3/2}}{15015 a^5 x^{5/2}}-\frac{\left (128 b^4 (10 A b-13 a B)\right ) \int \frac{\sqrt{a+b x}}{x^{5/2}} \, dx}{15015 a^5}\\ &=-\frac{2 A (a+b x)^{3/2}}{13 a x^{13/2}}+\frac{2 (10 A b-13 a B) (a+b x)^{3/2}}{143 a^2 x^{11/2}}-\frac{16 b (10 A b-13 a B) (a+b x)^{3/2}}{1287 a^3 x^{9/2}}+\frac{32 b^2 (10 A b-13 a B) (a+b x)^{3/2}}{3003 a^4 x^{7/2}}-\frac{128 b^3 (10 A b-13 a B) (a+b x)^{3/2}}{15015 a^5 x^{5/2}}+\frac{256 b^4 (10 A b-13 a B) (a+b x)^{3/2}}{45045 a^6 x^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0399784, size = 114, normalized size = 0.62 \[ -\frac{2 (a+b x)^{3/2} \left (80 a^3 b^2 x^2 (35 A+39 B x)-96 a^2 b^3 x^3 (25 A+26 B x)-70 a^4 b x (45 A+52 B x)+315 a^5 (11 A+13 B x)+128 a b^4 x^4 (15 A+13 B x)-1280 A b^5 x^5\right )}{45045 a^6 x^{13/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[a + b*x]*(A + B*x))/x^(15/2),x]

[Out]

(-2*(a + b*x)^(3/2)*(-1280*A*b^5*x^5 + 315*a^5*(11*A + 13*B*x) + 128*a*b^4*x^4*(15*A + 13*B*x) - 96*a^2*b^3*x^
3*(25*A + 26*B*x) + 80*a^3*b^2*x^2*(35*A + 39*B*x) - 70*a^4*b*x*(45*A + 52*B*x)))/(45045*a^6*x^(13/2))

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Maple [A]  time = 0.005, size = 125, normalized size = 0.7 \begin{align*} -{\frac{-2560\,A{b}^{5}{x}^{5}+3328\,B{x}^{5}a{b}^{4}+3840\,aA{b}^{4}{x}^{4}-4992\,B{x}^{4}{a}^{2}{b}^{3}-4800\,{a}^{2}A{b}^{3}{x}^{3}+6240\,B{x}^{3}{a}^{3}{b}^{2}+5600\,{a}^{3}A{b}^{2}{x}^{2}-7280\,B{x}^{2}{a}^{4}b-6300\,{a}^{4}Abx+8190\,{a}^{5}Bx+6930\,A{a}^{5}}{45045\,{a}^{6}} \left ( bx+a \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{13}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b*x+a)^(1/2)/x^(15/2),x)

[Out]

-2/45045*(b*x+a)^(3/2)*(-1280*A*b^5*x^5+1664*B*a*b^4*x^5+1920*A*a*b^4*x^4-2496*B*a^2*b^3*x^4-2400*A*a^2*b^3*x^
3+3120*B*a^3*b^2*x^3+2800*A*a^3*b^2*x^2-3640*B*a^4*b*x^2-3150*A*a^4*b*x+4095*B*a^5*x+3465*A*a^5)/x^(13/2)/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)*(b*x+a)^(1/2)/x^(15/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.64475, size = 360, normalized size = 1.97 \begin{align*} -\frac{2 \,{\left (3465 \, A a^{6} + 128 \,{\left (13 \, B a b^{5} - 10 \, A b^{6}\right )} x^{6} - 64 \,{\left (13 \, B a^{2} b^{4} - 10 \, A a b^{5}\right )} x^{5} + 48 \,{\left (13 \, B a^{3} b^{3} - 10 \, A a^{2} b^{4}\right )} x^{4} - 40 \,{\left (13 \, B a^{4} b^{2} - 10 \, A a^{3} b^{3}\right )} x^{3} + 35 \,{\left (13 \, B a^{5} b - 10 \, A a^{4} b^{2}\right )} x^{2} + 315 \,{\left (13 \, B a^{6} + A a^{5} b\right )} x\right )} \sqrt{b x + a}}{45045 \, a^{6} x^{\frac{13}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b*x+a)^(1/2)/x^(15/2),x, algorithm="fricas")

[Out]

-2/45045*(3465*A*a^6 + 128*(13*B*a*b^5 - 10*A*b^6)*x^6 - 64*(13*B*a^2*b^4 - 10*A*a*b^5)*x^5 + 48*(13*B*a^3*b^3
- 10*A*a^2*b^4)*x^4 - 40*(13*B*a^4*b^2 - 10*A*a^3*b^3)*x^3 + 35*(13*B*a^5*b - 10*A*a^4*b^2)*x^2 + 315*(13*B*a
^6 + A*a^5*b)*x)*sqrt(b*x + a)/(a^6*x^(13/2))

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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**(15/2),x)

[Out]

Timed out

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Giac [A]  time = 1.40972, 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 (13 \, B a b^{12} - 10 \, A b^{13}\right )}{\left (b x + a\right )}}{a^{7} b^{21}} - \frac{13 \,{\left (13 \, B a^{2} b^{12} - 10 \, A a b^{13}\right )}}{a^{7} b^{21}}\right )} + \frac{143 \,{\left (13 \, B a^{3} b^{12} - 10 \, A a^{2} b^{13}\right )}}{a^{7} b^{21}}\right )} - \frac{429 \,{\left (13 \, B a^{4} b^{12} - 10 \, A a^{3} b^{13}\right )}}{a^{7} b^{21}}\right )}{\left (b x + a\right )} + \frac{3003 \,{\left (13 \, B a^{5} b^{12} - 10 \, A a^{4} b^{13}\right )}}{a^{7} b^{21}}\right )}{\left (b x + a\right )} - \frac{15015 \,{\left (B a^{6} b^{12} - A a^{5} b^{13}\right )}}{a^{7} b^{21}}\right )}{\left (b x + a\right )}^{\frac{3}{2}} b}{33210777600 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{13}{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^(15/2),x, algorithm="giac")

[Out]

1/33210777600*((8*(2*(b*x + a)*(4*(b*x + a)*(2*(13*B*a*b^12 - 10*A*b^13)*(b*x + a)/(a^7*b^21) - 13*(13*B*a^2*b
^12 - 10*A*a*b^13)/(a^7*b^21)) + 143*(13*B*a^3*b^12 - 10*A*a^2*b^13)/(a^7*b^21)) - 429*(13*B*a^4*b^12 - 10*A*a
^3*b^13)/(a^7*b^21))*(b*x + a) + 3003*(13*B*a^5*b^12 - 10*A*a^4*b^13)/(a^7*b^21))*(b*x + a) - 15015*(B*a^6*b^1
2 - A*a^5*b^13)/(a^7*b^21))*(b*x + a)^(3/2)*b/(((b*x + a)*b - a*b)^(13/2)*abs(b))