Integrand size = 15, antiderivative size = 68 \[ \int \frac {(a+b x)^{5/2}}{x^{13/2}} \, dx=-\frac {2 (a+b x)^{7/2}}{11 a x^{11/2}}+\frac {8 b (a+b x)^{7/2}}{99 a^2 x^{9/2}}-\frac {16 b^2 (a+b x)^{7/2}}{693 a^3 x^{7/2}} \] Output:
-2/11*(b*x+a)^(7/2)/a/x^(11/2)+8/99*b*(b*x+a)^(7/2)/a^2/x^(9/2)-16/693*b^2 *(b*x+a)^(7/2)/a^3/x^(7/2)
Time = 0.12 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.59 \[ \int \frac {(a+b x)^{5/2}}{x^{13/2}} \, dx=-\frac {2 (a+b x)^{7/2} \left (63 a^2-28 a b x+8 b^2 x^2\right )}{693 a^3 x^{11/2}} \] Input:
Integrate[(a + b*x)^(5/2)/x^(13/2),x]
Output:
(-2*(a + b*x)^(7/2)*(63*a^2 - 28*a*b*x + 8*b^2*x^2))/(693*a^3*x^(11/2))
Time = 0.16 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {55, 55, 48}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(a+b x)^{5/2}}{x^{13/2}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {4 b \int \frac {(a+b x)^{5/2}}{x^{11/2}}dx}{11 a}-\frac {2 (a+b x)^{7/2}}{11 a x^{11/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {4 b \left (-\frac {2 b \int \frac {(a+b x)^{5/2}}{x^{9/2}}dx}{9 a}-\frac {2 (a+b x)^{7/2}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 (a+b x)^{7/2}}{11 a x^{11/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {4 b \left (\frac {4 b (a+b x)^{7/2}}{63 a^2 x^{7/2}}-\frac {2 (a+b x)^{7/2}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 (a+b x)^{7/2}}{11 a x^{11/2}}\) |
Input:
Int[(a + b*x)^(5/2)/x^(13/2),x]
Output:
(-2*(a + b*x)^(7/2))/(11*a*x^(11/2)) - (4*b*((-2*(a + b*x)^(7/2))/(9*a*x^( 9/2)) + (4*b*(a + b*x)^(7/2))/(63*a^2*x^(7/2))))/(11*a)
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] && EqQ[m + n + 2, 0] && NeQ[m, -1]
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] - Simp[d*(S implify[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] && 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] || !SumSimp lerQ[n, 1])
Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.51
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (8 b^{2} x^{2}-28 a b x +63 a^{2}\right )}{693 x^{\frac {11}{2}} a^{3}}\) | \(35\) |
orering | \(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (8 b^{2} x^{2}-28 a b x +63 a^{2}\right )}{693 x^{\frac {11}{2}} a^{3}}\) | \(35\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (8 b^{5} x^{5}-4 a \,b^{4} x^{4}+3 a^{2} b^{3} x^{3}+113 a^{3} b^{2} x^{2}+161 a^{4} b x +63 a^{5}\right )}{693 x^{\frac {11}{2}} a^{3}}\) | \(68\) |
default | \(-\frac {\left (b x +a \right )^{\frac {5}{2}}}{3 x^{\frac {11}{2}}}-\frac {5 a \left (-\frac {\left (b x +a \right )^{\frac {3}{2}}}{4 x^{\frac {11}{2}}}-\frac {3 a \left (-\frac {\sqrt {b x +a}}{5 x^{\frac {11}{2}}}-\frac {a \left (-\frac {2 \sqrt {b x +a}}{11 a \,x^{\frac {11}{2}}}-\frac {10 b \left (-\frac {2 \sqrt {b x +a}}{9 a \,x^{\frac {9}{2}}}-\frac {8 b \left (-\frac {2 \sqrt {b x +a}}{7 a \,x^{\frac {7}{2}}}-\frac {6 b \left (-\frac {2 \sqrt {b x +a}}{5 a \,x^{\frac {5}{2}}}-\frac {4 b \left (-\frac {2 \sqrt {b x +a}}{3 a \,x^{\frac {3}{2}}}+\frac {4 b \sqrt {b x +a}}{3 a^{2} \sqrt {x}}\right )}{5 a}\right )}{7 a}\right )}{9 a}\right )}{11 a}\right )}{10}\right )}{8}\right )}{6}\) | \(169\) |
Input:
int((b*x+a)^(5/2)/x^(13/2),x,method=_RETURNVERBOSE)
Output:
-2/693*(b*x+a)^(7/2)*(8*b^2*x^2-28*a*b*x+63*a^2)/x^(11/2)/a^3
Time = 0.07 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x)^{5/2}}{x^{13/2}} \, dx=-\frac {2 \, {\left (8 \, b^{5} x^{5} - 4 \, a b^{4} x^{4} + 3 \, a^{2} b^{3} x^{3} + 113 \, a^{3} b^{2} x^{2} + 161 \, a^{4} b x + 63 \, a^{5}\right )} \sqrt {b x + a}}{693 \, a^{3} x^{\frac {11}{2}}} \] Input:
integrate((b*x+a)^(5/2)/x^(13/2),x, algorithm="fricas")
Output:
-2/693*(8*b^5*x^5 - 4*a*b^4*x^4 + 3*a^2*b^3*x^3 + 113*a^3*b^2*x^2 + 161*a^ 4*b*x + 63*a^5)*sqrt(b*x + a)/(a^3*x^(11/2))
Leaf count of result is larger than twice the leaf count of optimal. 464 vs. \(2 (63) = 126\).
Time = 44.57 (sec) , antiderivative size = 464, normalized size of antiderivative = 6.82 \[ \int \frac {(a+b x)^{5/2}}{x^{13/2}} \, dx=- \frac {126 a^{7} b^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}}{x \left (693 a^{5} b^{4} x^{4} + 1386 a^{4} b^{5} x^{5} + 693 a^{3} b^{6} x^{6}\right )} - \frac {574 a^{6} b^{\frac {11}{2}} \sqrt {\frac {a}{b x} + 1}}{693 a^{5} b^{4} x^{4} + 1386 a^{4} b^{5} x^{5} + 693 a^{3} b^{6} x^{6}} - \frac {996 a^{5} b^{\frac {13}{2}} x \sqrt {\frac {a}{b x} + 1}}{693 a^{5} b^{4} x^{4} + 1386 a^{4} b^{5} x^{5} + 693 a^{3} b^{6} x^{6}} - \frac {780 a^{4} b^{\frac {15}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{693 a^{5} b^{4} x^{4} + 1386 a^{4} b^{5} x^{5} + 693 a^{3} b^{6} x^{6}} - \frac {230 a^{3} b^{\frac {17}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{693 a^{5} b^{4} x^{4} + 1386 a^{4} b^{5} x^{5} + 693 a^{3} b^{6} x^{6}} - \frac {6 a^{2} b^{\frac {19}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{693 a^{5} b^{4} x^{4} + 1386 a^{4} b^{5} x^{5} + 693 a^{3} b^{6} x^{6}} - \frac {24 a b^{\frac {21}{2}} x^{5} \sqrt {\frac {a}{b x} + 1}}{693 a^{5} b^{4} x^{4} + 1386 a^{4} b^{5} x^{5} + 693 a^{3} b^{6} x^{6}} - \frac {16 b^{\frac {23}{2}} x^{6} \sqrt {\frac {a}{b x} + 1}}{693 a^{5} b^{4} x^{4} + 1386 a^{4} b^{5} x^{5} + 693 a^{3} b^{6} x^{6}} \] Input:
integrate((b*x+a)**(5/2)/x**(13/2),x)
Output:
-126*a**7*b**(9/2)*sqrt(a/(b*x) + 1)/(x*(693*a**5*b**4*x**4 + 1386*a**4*b* *5*x**5 + 693*a**3*b**6*x**6)) - 574*a**6*b**(11/2)*sqrt(a/(b*x) + 1)/(693 *a**5*b**4*x**4 + 1386*a**4*b**5*x**5 + 693*a**3*b**6*x**6) - 996*a**5*b** (13/2)*x*sqrt(a/(b*x) + 1)/(693*a**5*b**4*x**4 + 1386*a**4*b**5*x**5 + 693 *a**3*b**6*x**6) - 780*a**4*b**(15/2)*x**2*sqrt(a/(b*x) + 1)/(693*a**5*b** 4*x**4 + 1386*a**4*b**5*x**5 + 693*a**3*b**6*x**6) - 230*a**3*b**(17/2)*x* *3*sqrt(a/(b*x) + 1)/(693*a**5*b**4*x**4 + 1386*a**4*b**5*x**5 + 693*a**3* b**6*x**6) - 6*a**2*b**(19/2)*x**4*sqrt(a/(b*x) + 1)/(693*a**5*b**4*x**4 + 1386*a**4*b**5*x**5 + 693*a**3*b**6*x**6) - 24*a*b**(21/2)*x**5*sqrt(a/(b *x) + 1)/(693*a**5*b**4*x**4 + 1386*a**4*b**5*x**5 + 693*a**3*b**6*x**6) - 16*b**(23/2)*x**6*sqrt(a/(b*x) + 1)/(693*a**5*b**4*x**4 + 1386*a**4*b**5* x**5 + 693*a**3*b**6*x**6)
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.68 \[ \int \frac {(a+b x)^{5/2}}{x^{13/2}} \, dx=-\frac {2 \, {\left (\frac {99 \, {\left (b x + a\right )}^{\frac {7}{2}} b^{2}}{x^{\frac {7}{2}}} - \frac {154 \, {\left (b x + a\right )}^{\frac {9}{2}} b}{x^{\frac {9}{2}}} + \frac {63 \, {\left (b x + a\right )}^{\frac {11}{2}}}{x^{\frac {11}{2}}}\right )}}{693 \, a^{3}} \] Input:
integrate((b*x+a)^(5/2)/x^(13/2),x, algorithm="maxima")
Output:
-2/693*(99*(b*x + a)^(7/2)*b^2/x^(7/2) - 154*(b*x + a)^(9/2)*b/x^(9/2) + 6 3*(b*x + a)^(11/2)/x^(11/2))/a^3
Time = 0.14 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^{5/2}}{x^{13/2}} \, dx=-\frac {2 \, {\left (\frac {99 \, b^{5}}{a} + 4 \, {\left (\frac {2 \, {\left (b x + a\right )} b^{5}}{a^{3}} - \frac {11 \, b^{5}}{a^{2}}\right )} {\left (b x + a\right )}\right )} {\left (b x + a\right )}^{\frac {7}{2}} b^{7}}{693 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {11}{2}} {\left | b \right |}} \] Input:
integrate((b*x+a)^(5/2)/x^(13/2),x, algorithm="giac")
Output:
-2/693*(99*b^5/a + 4*(2*(b*x + a)*b^5/a^3 - 11*b^5/a^2)*(b*x + a))*(b*x + a)^(7/2)*b^7/(((b*x + a)*b - a*b)^(11/2)*abs(b))
Time = 0.21 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x)^{5/2}}{x^{13/2}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,a^2}{11}+\frac {226\,b^2\,x^2}{693}+\frac {46\,a\,b\,x}{99}+\frac {2\,b^3\,x^3}{231\,a}-\frac {8\,b^4\,x^4}{693\,a^2}+\frac {16\,b^5\,x^5}{693\,a^3}\right )}{x^{11/2}} \] Input:
int((a + b*x)^(5/2)/x^(13/2),x)
Output:
-((a + b*x)^(1/2)*((2*a^2)/11 + (226*b^2*x^2)/693 + (46*a*b*x)/99 + (2*b^3 *x^3)/(231*a) - (8*b^4*x^4)/(693*a^2) + (16*b^5*x^5)/(693*a^3)))/x^(11/2)
Time = 0.17 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.74 \[ \int \frac {(a+b x)^{5/2}}{x^{13/2}} \, dx=\frac {-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{5}}{11}-\frac {46 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b x}{99}-\frac {226 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{2} x^{2}}{693}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{3} x^{3}}{231}+\frac {8 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{4} x^{4}}{693}-\frac {16 \sqrt {x}\, \sqrt {b x +a}\, b^{5} x^{5}}{693}+\frac {16 \sqrt {b}\, b^{5} x^{6}}{693}}{a^{3} x^{6}} \] Input:
int((b*x+a)^(5/2)/x^(13/2),x)
Output:
(2*( - 63*sqrt(x)*sqrt(a + b*x)*a**5 - 161*sqrt(x)*sqrt(a + b*x)*a**4*b*x - 113*sqrt(x)*sqrt(a + b*x)*a**3*b**2*x**2 - 3*sqrt(x)*sqrt(a + b*x)*a**2* b**3*x**3 + 4*sqrt(x)*sqrt(a + b*x)*a*b**4*x**4 - 8*sqrt(x)*sqrt(a + b*x)* b**5*x**5 + 8*sqrt(b)*b**5*x**6))/(693*a**3*x**6)