3.511 \(\int \frac {(a+b x)^{5/2} (A+B x)}{x^{15/2}} \, dx\)

Optimal. Leaf size=117 \[ \frac {16 b^2 (a+b x)^{7/2} (6 A b-13 a B)}{9009 a^4 x^{7/2}}-\frac {8 b (a+b x)^{7/2} (6 A b-13 a B)}{1287 a^3 x^{9/2}}+\frac {2 (a+b x)^{7/2} (6 A b-13 a B)}{143 a^2 x^{11/2}}-\frac {2 A (a+b x)^{7/2}}{13 a x^{13/2}} \]

[Out]

-2/13*A*(b*x+a)^(7/2)/a/x^(13/2)+2/143*(6*A*b-13*B*a)*(b*x+a)^(7/2)/a^2/x^(11/2)-8/1287*b*(6*A*b-13*B*a)*(b*x+
a)^(7/2)/a^3/x^(9/2)+16/9009*b^2*(6*A*b-13*B*a)*(b*x+a)^(7/2)/a^4/x^(7/2)

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {78, 45, 37} \[ \frac {16 b^2 (a+b x)^{7/2} (6 A b-13 a B)}{9009 a^4 x^{7/2}}-\frac {8 b (a+b x)^{7/2} (6 A b-13 a B)}{1287 a^3 x^{9/2}}+\frac {2 (a+b x)^{7/2} (6 A b-13 a B)}{143 a^2 x^{11/2}}-\frac {2 A (a+b x)^{7/2}}{13 a x^{13/2}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(5/2)*(A + B*x))/x^(15/2),x]

[Out]

(-2*A*(a + b*x)^(7/2))/(13*a*x^(13/2)) + (2*(6*A*b - 13*a*B)*(a + b*x)^(7/2))/(143*a^2*x^(11/2)) - (8*b*(6*A*b
- 13*a*B)*(a + b*x)^(7/2))/(1287*a^3*x^(9/2)) + (16*b^2*(6*A*b - 13*a*B)*(a + b*x)^(7/2))/(9009*a^4*x^(7/2))

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]

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 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]))))

Rubi steps

\begin {align*} \int \frac {(a+b x)^{5/2} (A+B x)}{x^{15/2}} \, dx &=-\frac {2 A (a+b x)^{7/2}}{13 a x^{13/2}}+\frac {\left (2 \left (-3 A b+\frac {13 a B}{2}\right )\right ) \int \frac {(a+b x)^{5/2}}{x^{13/2}} \, dx}{13 a}\\ &=-\frac {2 A (a+b x)^{7/2}}{13 a x^{13/2}}+\frac {2 (6 A b-13 a B) (a+b x)^{7/2}}{143 a^2 x^{11/2}}+\frac {(4 b (6 A b-13 a B)) \int \frac {(a+b x)^{5/2}}{x^{11/2}} \, dx}{143 a^2}\\ &=-\frac {2 A (a+b x)^{7/2}}{13 a x^{13/2}}+\frac {2 (6 A b-13 a B) (a+b x)^{7/2}}{143 a^2 x^{11/2}}-\frac {8 b (6 A b-13 a B) (a+b x)^{7/2}}{1287 a^3 x^{9/2}}-\frac {\left (8 b^2 (6 A b-13 a B)\right ) \int \frac {(a+b x)^{5/2}}{x^{9/2}} \, dx}{1287 a^3}\\ &=-\frac {2 A (a+b x)^{7/2}}{13 a x^{13/2}}+\frac {2 (6 A b-13 a B) (a+b x)^{7/2}}{143 a^2 x^{11/2}}-\frac {8 b (6 A b-13 a B) (a+b x)^{7/2}}{1287 a^3 x^{9/2}}+\frac {16 b^2 (6 A b-13 a B) (a+b x)^{7/2}}{9009 a^4 x^{7/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 76, normalized size = 0.65 \[ -\frac {2 (a+b x)^{7/2} \left (63 a^3 (11 A+13 B x)-14 a^2 b x (27 A+26 B x)+8 a b^2 x^2 (21 A+13 B x)-48 A b^3 x^3\right )}{9009 a^4 x^{13/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(5/2)*(A + B*x))/x^(15/2),x]

[Out]

(-2*(a + b*x)^(7/2)*(-48*A*b^3*x^3 + 63*a^3*(11*A + 13*B*x) + 8*a*b^2*x^2*(21*A + 13*B*x) - 14*a^2*b*x*(27*A +
26*B*x)))/(9009*a^4*x^(13/2))

________________________________________________________________________________________

fricas [A]  time = 0.55, size = 149, normalized size = 1.27 \[ -\frac {2 \, {\left (693 \, A a^{6} + 8 \, {\left (13 \, B a b^{5} - 6 \, A b^{6}\right )} x^{6} - 4 \, {\left (13 \, B a^{2} b^{4} - 6 \, A a b^{5}\right )} x^{5} + 3 \, {\left (13 \, B a^{3} b^{3} - 6 \, A a^{2} b^{4}\right )} x^{4} + {\left (1469 \, B a^{4} b^{2} + 15 \, A a^{3} b^{3}\right )} x^{3} + 7 \, {\left (299 \, B a^{5} b + 159 \, A a^{4} b^{2}\right )} x^{2} + 63 \, {\left (13 \, B a^{6} + 27 \, A a^{5} b\right )} x\right )} \sqrt {b x + a}}{9009 \, a^{4} x^{\frac {13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/x^(15/2),x, algorithm="fricas")

[Out]

-2/9009*(693*A*a^6 + 8*(13*B*a*b^5 - 6*A*b^6)*x^6 - 4*(13*B*a^2*b^4 - 6*A*a*b^5)*x^5 + 3*(13*B*a^3*b^3 - 6*A*a
^2*b^4)*x^4 + (1469*B*a^4*b^2 + 15*A*a^3*b^3)*x^3 + 7*(299*B*a^5*b + 159*A*a^4*b^2)*x^2 + 63*(13*B*a^6 + 27*A*
a^5*b)*x)*sqrt(b*x + a)/(a^4*x^(13/2))

________________________________________________________________________________________

giac [A]  time = 1.22, size = 144, normalized size = 1.23 \[ -\frac {2 \, {\left ({\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (13 \, B a^{3} b^{12} - 6 \, A a^{2} b^{13}\right )} {\left (b x + a\right )}}{a^{6}} - \frac {13 \, {\left (13 \, B a^{4} b^{12} - 6 \, A a^{3} b^{13}\right )}}{a^{6}}\right )} + \frac {143 \, {\left (13 \, B a^{5} b^{12} - 6 \, A a^{4} b^{13}\right )}}{a^{6}}\right )} - \frac {1287 \, {\left (B a^{6} b^{12} - A a^{5} b^{13}\right )}}{a^{6}}\right )} {\left (b x + a\right )}^{\frac {7}{2}} b}{9009 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {13}{2}} {\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/x^(15/2),x, algorithm="giac")

[Out]

-2/9009*((b*x + a)*(4*(b*x + a)*(2*(13*B*a^3*b^12 - 6*A*a^2*b^13)*(b*x + a)/a^6 - 13*(13*B*a^4*b^12 - 6*A*a^3*
b^13)/a^6) + 143*(13*B*a^5*b^12 - 6*A*a^4*b^13)/a^6) - 1287*(B*a^6*b^12 - A*a^5*b^13)/a^6)*(b*x + a)^(7/2)*b/(
((b*x + a)*b - a*b)^(13/2)*abs(b))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 77, normalized size = 0.66 \[ -\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-48 A \,b^{3} x^{3}+104 B a \,b^{2} x^{3}+168 A a \,b^{2} x^{2}-364 B \,a^{2} b \,x^{2}-378 A \,a^{2} b x +819 B \,a^{3} x +693 A \,a^{3}\right )}{9009 a^{4} x^{\frac {13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/2)*(B*x+A)/x^(15/2),x)

[Out]

-2/9009*(b*x+a)^(7/2)*(-48*A*b^3*x^3+104*B*a*b^2*x^3+168*A*a*b^2*x^2-364*B*a^2*b*x^2-378*A*a^2*b*x+819*B*a^3*x
+693*A*a^3)/x^(13/2)/a^4

________________________________________________________________________________________

maxima [B]  time = 1.03, size = 350, normalized size = 2.99 \[ -\frac {16 \, \sqrt {b x^{2} + a x} B b^{5}}{693 \, a^{3} x} + \frac {32 \, \sqrt {b x^{2} + a x} A b^{6}}{3003 \, a^{4} x} + \frac {8 \, \sqrt {b x^{2} + a x} B b^{4}}{693 \, a^{2} x^{2}} - \frac {16 \, \sqrt {b x^{2} + a x} A b^{5}}{3003 \, a^{3} x^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} B b^{3}}{231 \, a x^{3}} + \frac {4 \, \sqrt {b x^{2} + a x} A b^{4}}{1001 \, a^{2} x^{3}} + \frac {5 \, \sqrt {b x^{2} + a x} B b^{2}}{693 \, x^{4}} - \frac {10 \, \sqrt {b x^{2} + a x} A b^{3}}{3003 \, a x^{4}} - \frac {5 \, \sqrt {b x^{2} + a x} B a b}{792 \, x^{5}} + \frac {5 \, \sqrt {b x^{2} + a x} A b^{2}}{1716 \, x^{5}} - \frac {5 \, \sqrt {b x^{2} + a x} B a^{2}}{88 \, x^{6}} - \frac {3 \, \sqrt {b x^{2} + a x} A a b}{1144 \, x^{6}} + \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a}{24 \, x^{7}} - \frac {3 \, \sqrt {b x^{2} + a x} A a^{2}}{104 \, x^{7}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} B}{3 \, x^{8}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a}{8 \, x^{8}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} A}{4 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/x^(15/2),x, algorithm="maxima")

[Out]

-16/693*sqrt(b*x^2 + a*x)*B*b^5/(a^3*x) + 32/3003*sqrt(b*x^2 + a*x)*A*b^6/(a^4*x) + 8/693*sqrt(b*x^2 + a*x)*B*
b^4/(a^2*x^2) - 16/3003*sqrt(b*x^2 + a*x)*A*b^5/(a^3*x^2) - 2/231*sqrt(b*x^2 + a*x)*B*b^3/(a*x^3) + 4/1001*sqr
t(b*x^2 + a*x)*A*b^4/(a^2*x^3) + 5/693*sqrt(b*x^2 + a*x)*B*b^2/x^4 - 10/3003*sqrt(b*x^2 + a*x)*A*b^3/(a*x^4) -
5/792*sqrt(b*x^2 + a*x)*B*a*b/x^5 + 5/1716*sqrt(b*x^2 + a*x)*A*b^2/x^5 - 5/88*sqrt(b*x^2 + a*x)*B*a^2/x^6 - 3
/1144*sqrt(b*x^2 + a*x)*A*a*b/x^6 + 5/24*(b*x^2 + a*x)^(3/2)*B*a/x^7 - 3/104*sqrt(b*x^2 + a*x)*A*a^2/x^7 - 1/3
*(b*x^2 + a*x)^(5/2)*B/x^8 + 1/8*(b*x^2 + a*x)^(3/2)*A*a/x^8 - 1/4*(b*x^2 + a*x)^(5/2)*A/x^9

________________________________________________________________________________________

mupad [B]  time = 0.95, size = 136, normalized size = 1.16 \[ -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A\,a^2}{13}+\frac {x\,\left (1638\,B\,a^6+3402\,A\,b\,a^5\right )}{9009\,a^4}-\frac {x^6\,\left (96\,A\,b^6-208\,B\,a\,b^5\right )}{9009\,a^4}+\frac {2\,b\,x^2\,\left (159\,A\,b+299\,B\,a\right )}{1287}-\frac {2\,b^3\,x^4\,\left (6\,A\,b-13\,B\,a\right )}{3003\,a^2}+\frac {8\,b^4\,x^5\,\left (6\,A\,b-13\,B\,a\right )}{9009\,a^3}+\frac {2\,b^2\,x^3\,\left (15\,A\,b+1469\,B\,a\right )}{9009\,a}\right )}{x^{13/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^(5/2))/x^(15/2),x)

[Out]

-((a + b*x)^(1/2)*((2*A*a^2)/13 + (x*(1638*B*a^6 + 3402*A*a^5*b))/(9009*a^4) - (x^6*(96*A*b^6 - 208*B*a*b^5))/
(9009*a^4) + (2*b*x^2*(159*A*b + 299*B*a))/1287 - (2*b^3*x^4*(6*A*b - 13*B*a))/(3003*a^2) + (8*b^4*x^5*(6*A*b
- 13*B*a))/(9009*a^3) + (2*b^2*x^3*(15*A*b + 1469*B*a))/(9009*a)))/x^(13/2)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(5/2)*(B*x+A)/x**(15/2),x)

[Out]

Timed out

________________________________________________________________________________________