3.512 \(\int \frac{(a+b x)^{5/2} (A+B x)}{x^{17/2}} \, dx\)

Optimal. Leaf size=150 \[ -\frac{32 b^3 (a+b x)^{7/2} (8 A b-15 a B)}{45045 a^5 x^{7/2}}+\frac{16 b^2 (a+b x)^{7/2} (8 A b-15 a B)}{6435 a^4 x^{9/2}}-\frac{4 b (a+b x)^{7/2} (8 A b-15 a B)}{715 a^3 x^{11/2}}+\frac{2 (a+b x)^{7/2} (8 A b-15 a B)}{195 a^2 x^{13/2}}-\frac{2 A (a+b x)^{7/2}}{15 a x^{15/2}} \]

[Out]

(-2*A*(a + b*x)^(7/2))/(15*a*x^(15/2)) + (2*(8*A*b - 15*a*B)*(a + b*x)^(7/2))/(195*a^2*x^(13/2)) - (4*b*(8*A*b
- 15*a*B)*(a + b*x)^(7/2))/(715*a^3*x^(11/2)) + (16*b^2*(8*A*b - 15*a*B)*(a + b*x)^(7/2))/(6435*a^4*x^(9/2))
- (32*b^3*(8*A*b - 15*a*B)*(a + b*x)^(7/2))/(45045*a^5*x^(7/2))

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Rubi [A]  time = 0.0540292, 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} \[ -\frac{32 b^3 (a+b x)^{7/2} (8 A b-15 a B)}{45045 a^5 x^{7/2}}+\frac{16 b^2 (a+b x)^{7/2} (8 A b-15 a B)}{6435 a^4 x^{9/2}}-\frac{4 b (a+b x)^{7/2} (8 A b-15 a B)}{715 a^3 x^{11/2}}+\frac{2 (a+b x)^{7/2} (8 A b-15 a B)}{195 a^2 x^{13/2}}-\frac{2 A (a+b x)^{7/2}}{15 a x^{15/2}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(5/2)*(A + B*x))/x^(17/2),x]

[Out]

(-2*A*(a + b*x)^(7/2))/(15*a*x^(15/2)) + (2*(8*A*b - 15*a*B)*(a + b*x)^(7/2))/(195*a^2*x^(13/2)) - (4*b*(8*A*b
- 15*a*B)*(a + b*x)^(7/2))/(715*a^3*x^(11/2)) + (16*b^2*(8*A*b - 15*a*B)*(a + b*x)^(7/2))/(6435*a^4*x^(9/2))
- (32*b^3*(8*A*b - 15*a*B)*(a + b*x)^(7/2))/(45045*a^5*x^(7/2))

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{(a+b x)^{5/2} (A+B x)}{x^{17/2}} \, dx &=-\frac{2 A (a+b x)^{7/2}}{15 a x^{15/2}}+\frac{\left (2 \left (-4 A b+\frac{15 a B}{2}\right )\right ) \int \frac{(a+b x)^{5/2}}{x^{15/2}} \, dx}{15 a}\\ &=-\frac{2 A (a+b x)^{7/2}}{15 a x^{15/2}}+\frac{2 (8 A b-15 a B) (a+b x)^{7/2}}{195 a^2 x^{13/2}}+\frac{(2 b (8 A b-15 a B)) \int \frac{(a+b x)^{5/2}}{x^{13/2}} \, dx}{65 a^2}\\ &=-\frac{2 A (a+b x)^{7/2}}{15 a x^{15/2}}+\frac{2 (8 A b-15 a B) (a+b x)^{7/2}}{195 a^2 x^{13/2}}-\frac{4 b (8 A b-15 a B) (a+b x)^{7/2}}{715 a^3 x^{11/2}}-\frac{\left (8 b^2 (8 A b-15 a B)\right ) \int \frac{(a+b x)^{5/2}}{x^{11/2}} \, dx}{715 a^3}\\ &=-\frac{2 A (a+b x)^{7/2}}{15 a x^{15/2}}+\frac{2 (8 A b-15 a B) (a+b x)^{7/2}}{195 a^2 x^{13/2}}-\frac{4 b (8 A b-15 a B) (a+b x)^{7/2}}{715 a^3 x^{11/2}}+\frac{16 b^2 (8 A b-15 a B) (a+b x)^{7/2}}{6435 a^4 x^{9/2}}+\frac{\left (16 b^3 (8 A b-15 a B)\right ) \int \frac{(a+b x)^{5/2}}{x^{9/2}} \, dx}{6435 a^4}\\ &=-\frac{2 A (a+b x)^{7/2}}{15 a x^{15/2}}+\frac{2 (8 A b-15 a B) (a+b x)^{7/2}}{195 a^2 x^{13/2}}-\frac{4 b (8 A b-15 a B) (a+b x)^{7/2}}{715 a^3 x^{11/2}}+\frac{16 b^2 (8 A b-15 a B) (a+b x)^{7/2}}{6435 a^4 x^{9/2}}-\frac{32 b^3 (8 A b-15 a B) (a+b x)^{7/2}}{45045 a^5 x^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0408002, size = 95, normalized size = 0.63 \[ -\frac{2 (a+b x)^{7/2} \left (168 a^2 b^2 x^2 (6 A+5 B x)-42 a^3 b x (44 A+45 B x)+231 a^4 (13 A+15 B x)-16 a b^3 x^3 (28 A+15 B x)+128 A b^4 x^4\right )}{45045 a^5 x^{15/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(5/2)*(A + B*x))/x^(17/2),x]

[Out]

(-2*(a + b*x)^(7/2)*(128*A*b^4*x^4 + 168*a^2*b^2*x^2*(6*A + 5*B*x) + 231*a^4*(13*A + 15*B*x) - 16*a*b^3*x^3*(2
8*A + 15*B*x) - 42*a^3*b*x*(44*A + 45*B*x)))/(45045*a^5*x^(15/2))

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Maple [A]  time = 0.004, size = 101, normalized size = 0.7 \begin{align*} -{\frac{256\,A{b}^{4}{x}^{4}-480\,Ba{b}^{3}{x}^{4}-896\,Aa{b}^{3}{x}^{3}+1680\,B{a}^{2}{b}^{2}{x}^{3}+2016\,A{a}^{2}{b}^{2}{x}^{2}-3780\,B{a}^{3}b{x}^{2}-3696\,A{a}^{3}bx+6930\,B{a}^{4}x+6006\,A{a}^{4}}{45045\,{a}^{5}} \left ( bx+a \right ) ^{{\frac{7}{2}}}{x}^{-{\frac{15}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/2)*(B*x+A)/x^(17/2),x)

[Out]

-2/45045*(b*x+a)^(7/2)*(128*A*b^4*x^4-240*B*a*b^3*x^4-448*A*a*b^3*x^3+840*B*a^2*b^2*x^3+1008*A*a^2*b^2*x^2-189
0*B*a^3*b*x^2-1848*A*a^3*b*x+3465*B*a^4*x+3003*A*a^4)/x^(15/2)/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)^(5/2)*(B*x+A)/x^(17/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.55256, size = 405, normalized size = 2.7 \begin{align*} -\frac{2 \,{\left (3003 \, A a^{7} - 16 \,{\left (15 \, B a b^{6} - 8 \, A b^{7}\right )} x^{7} + 8 \,{\left (15 \, B a^{2} b^{5} - 8 \, A a b^{6}\right )} x^{6} - 6 \,{\left (15 \, B a^{3} b^{4} - 8 \, A a^{2} b^{5}\right )} x^{5} + 5 \,{\left (15 \, B a^{4} b^{3} - 8 \, A a^{3} b^{4}\right )} x^{4} + 35 \,{\left (159 \, B a^{5} b^{2} + A a^{4} b^{3}\right )} x^{3} + 63 \,{\left (135 \, B a^{6} b + 71 \, A a^{5} b^{2}\right )} x^{2} + 231 \,{\left (15 \, B a^{7} + 31 \, A a^{6} b\right )} x\right )} \sqrt{b x + a}}{45045 \, a^{5} x^{\frac{15}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/x^(17/2),x, algorithm="fricas")

[Out]

-2/45045*(3003*A*a^7 - 16*(15*B*a*b^6 - 8*A*b^7)*x^7 + 8*(15*B*a^2*b^5 - 8*A*a*b^6)*x^6 - 6*(15*B*a^3*b^4 - 8*
A*a^2*b^5)*x^5 + 5*(15*B*a^4*b^3 - 8*A*a^3*b^4)*x^4 + 35*(159*B*a^5*b^2 + A*a^4*b^3)*x^3 + 63*(135*B*a^6*b + 7
1*A*a^5*b^2)*x^2 + 231*(15*B*a^7 + 31*A*a^6*b)*x)*sqrt(b*x + a)/(a^5*x^(15/2))

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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)**(5/2)*(B*x+A)/x**(17/2),x)

[Out]

Timed out

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Giac [A]  time = 2.06121, size = 258, normalized size = 1.72 \begin{align*} -\frac{{\left ({\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (15 \, B a^{3} b^{14} - 8 \, A a^{2} b^{15}\right )}{\left (b x + a\right )}}{a^{8} b^{24}} - \frac{15 \,{\left (15 \, B a^{4} b^{14} - 8 \, A a^{3} b^{15}\right )}}{a^{8} b^{24}}\right )} + \frac{195 \,{\left (15 \, B a^{5} b^{14} - 8 \, A a^{4} b^{15}\right )}}{a^{8} b^{24}}\right )} - \frac{715 \,{\left (15 \, B a^{6} b^{14} - 8 \, A a^{5} b^{15}\right )}}{a^{8} b^{24}}\right )}{\left (b x + a\right )} + \frac{6435 \,{\left (B a^{7} b^{14} - A a^{6} b^{15}\right )}}{a^{8} b^{24}}\right )}{\left (b x + a\right )}^{\frac{7}{2}} b}{2952069120 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{15}{2}}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/x^(17/2),x, algorithm="giac")

[Out]

-1/2952069120*((2*(b*x + a)*(4*(b*x + a)*(2*(15*B*a^3*b^14 - 8*A*a^2*b^15)*(b*x + a)/(a^8*b^24) - 15*(15*B*a^4
*b^14 - 8*A*a^3*b^15)/(a^8*b^24)) + 195*(15*B*a^5*b^14 - 8*A*a^4*b^15)/(a^8*b^24)) - 715*(15*B*a^6*b^14 - 8*A*
a^5*b^15)/(a^8*b^24))*(b*x + a) + 6435*(B*a^7*b^14 - A*a^6*b^15)/(a^8*b^24))*(b*x + a)^(7/2)*b/(((b*x + a)*b -
a*b)^(15/2)*abs(b))