Integrand size = 17, antiderivative size = 74 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^7} \, dx=-\frac {2 \left (a x+b x^2\right )^{5/2}}{9 a x^7}+\frac {8 b \left (a x+b x^2\right )^{5/2}}{63 a^2 x^6}-\frac {16 b^2 \left (a x+b x^2\right )^{5/2}}{315 a^3 x^5} \] Output:
-2/9*(b*x^2+a*x)^(5/2)/a/x^7+8/63*b*(b*x^2+a*x)^(5/2)/a^2/x^6-16/315*b^2*( b*x^2+a*x)^(5/2)/a^3/x^5
Time = 0.13 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.54 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^7} \, dx=-\frac {2 (x (a+b x))^{5/2} \left (35 a^2-20 a b x+8 b^2 x^2\right )}{315 a^3 x^7} \] Input:
Integrate[(a*x + b*x^2)^(3/2)/x^7,x]
Output:
(-2*(x*(a + b*x))^(5/2)*(35*a^2 - 20*a*b*x + 8*b^2*x^2))/(315*a^3*x^7)
Time = 0.33 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1129, 1129, 1123}
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.
\(\displaystyle \int \frac {\left (a x+b x^2\right )^{3/2}}{x^7} \, dx\) |
\(\Big \downarrow \) 1129 |
\(\displaystyle -\frac {4 b \int \frac {\left (b x^2+a x\right )^{3/2}}{x^6}dx}{9 a}-\frac {2 \left (a x+b x^2\right )^{5/2}}{9 a x^7}\) |
\(\Big \downarrow \) 1129 |
\(\displaystyle -\frac {4 b \left (-\frac {2 b \int \frac {\left (b x^2+a x\right )^{3/2}}{x^5}dx}{7 a}-\frac {2 \left (a x+b x^2\right )^{5/2}}{7 a x^6}\right )}{9 a}-\frac {2 \left (a x+b x^2\right )^{5/2}}{9 a x^7}\) |
\(\Big \downarrow \) 1123 |
\(\displaystyle -\frac {4 b \left (\frac {4 b \left (a x+b x^2\right )^{5/2}}{35 a^2 x^5}-\frac {2 \left (a x+b x^2\right )^{5/2}}{7 a x^6}\right )}{9 a}-\frac {2 \left (a x+b x^2\right )^{5/2}}{9 a x^7}\) |
Input:
Int[(a*x + b*x^2)^(3/2)/x^7,x]
Output:
(-2*(a*x + b*x^2)^(5/2))/(9*a*x^7) - (4*b*((-2*(a*x + b*x^2)^(5/2))/(7*a*x ^6) + (4*b*(a*x + b*x^2)^(5/2))/(35*a^2*x^5)))/(9*a)
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(2*c*d - b *e))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + 2*p + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* c*d - b*e))), x] + Simp[c*(Simplify[m + 2*p + 2]/((m + p + 1)*(2*c*d - b*e) )) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d , e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 2], 0]
Time = 0.40 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.57
method | result | size |
pseudoelliptic | \(-\frac {2 \sqrt {x \left (b x +a \right )}\, \left (b x +a \right )^{2} \left (\frac {8}{35} b^{2} x^{2}-\frac {4}{7} a b x +a^{2}\right )}{9 x^{5} a^{3}}\) | \(42\) |
gosper | \(-\frac {2 \left (b x +a \right ) \left (8 b^{2} x^{2}-20 a b x +35 a^{2}\right ) \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{315 x^{6} a^{3}}\) | \(44\) |
orering | \(-\frac {2 \left (b x +a \right ) \left (8 b^{2} x^{2}-20 a b x +35 a^{2}\right ) \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{315 x^{6} a^{3}}\) | \(44\) |
trager | \(-\frac {2 \left (8 b^{4} x^{4}-4 a \,b^{3} x^{3}+3 a^{2} b^{2} x^{2}+50 a^{3} b x +35 a^{4}\right ) \sqrt {b \,x^{2}+a x}}{315 a^{3} x^{5}}\) | \(61\) |
risch | \(-\frac {2 \left (b x +a \right ) \left (8 b^{4} x^{4}-4 a \,b^{3} x^{3}+3 a^{2} b^{2} x^{2}+50 a^{3} b x +35 a^{4}\right )}{315 x^{4} \sqrt {x \left (b x +a \right )}\, a^{3}}\) | \(64\) |
default | \(-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{9 a \,x^{7}}-\frac {4 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{7 a \,x^{6}}+\frac {4 b \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{35 a^{2} x^{5}}\right )}{9 a}\) | \(67\) |
Input:
int((b*x^2+a*x)^(3/2)/x^7,x,method=_RETURNVERBOSE)
Output:
-2/9*(x*(b*x+a))^(1/2)*(b*x+a)^2*(8/35*b^2*x^2-4/7*a*b*x+a^2)/x^5/a^3
Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^7} \, dx=-\frac {2 \, {\left (8 \, b^{4} x^{4} - 4 \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} + 50 \, a^{3} b x + 35 \, a^{4}\right )} \sqrt {b x^{2} + a x}}{315 \, a^{3} x^{5}} \] Input:
integrate((b*x^2+a*x)^(3/2)/x^7,x, algorithm="fricas")
Output:
-2/315*(8*b^4*x^4 - 4*a*b^3*x^3 + 3*a^2*b^2*x^2 + 50*a^3*b*x + 35*a^4)*sqr t(b*x^2 + a*x)/(a^3*x^5)
\[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^7} \, dx=\int \frac {\left (x \left (a + b x\right )\right )^{\frac {3}{2}}}{x^{7}}\, dx \] Input:
integrate((b*x**2+a*x)**(3/2)/x**7,x)
Output:
Integral((x*(a + b*x))**(3/2)/x**7, x)
Time = 0.03 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.58 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^7} \, dx=-\frac {16 \, \sqrt {b x^{2} + a x} b^{4}}{315 \, a^{3} x} + \frac {8 \, \sqrt {b x^{2} + a x} b^{3}}{315 \, a^{2} x^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} b^{2}}{105 \, a x^{3}} + \frac {\sqrt {b x^{2} + a x} b}{63 \, x^{4}} + \frac {\sqrt {b x^{2} + a x} a}{9 \, x^{5}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}}}{3 \, x^{6}} \] Input:
integrate((b*x^2+a*x)^(3/2)/x^7,x, algorithm="maxima")
Output:
-16/315*sqrt(b*x^2 + a*x)*b^4/(a^3*x) + 8/315*sqrt(b*x^2 + a*x)*b^3/(a^2*x ^2) - 2/105*sqrt(b*x^2 + a*x)*b^2/(a*x^3) + 1/63*sqrt(b*x^2 + a*x)*b/x^4 + 1/9*sqrt(b*x^2 + a*x)*a/x^5 - 1/3*(b*x^2 + a*x)^(3/2)/x^6
Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (62) = 124\).
Time = 0.28 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.62 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^7} \, dx=\frac {2 \, {\left (420 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{6} b^{3} + 1575 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} a b^{\frac {5}{2}} + 2583 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a^{2} b^{2} + 2310 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{3} b^{\frac {3}{2}} + 1170 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{4} b + 315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{5} \sqrt {b} + 35 \, a^{6}\right )}}{315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{9}} \] Input:
integrate((b*x^2+a*x)^(3/2)/x^7,x, algorithm="giac")
Output:
2/315*(420*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*b^3 + 1575*(sqrt(b)*x - sqrt( b*x^2 + a*x))^5*a*b^(5/2) + 2583*(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*a^2*b^2 + 2310*(sqrt(b)*x - sqrt(b*x^2 + a*x))^3*a^3*b^(3/2) + 1170*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a^4*b + 315*(sqrt(b)*x - sqrt(b*x^2 + a*x))*a^5*sqrt( b) + 35*a^6)/(sqrt(b)*x - sqrt(b*x^2 + a*x))^9
Time = 10.23 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^7} \, dx=\frac {8\,b^3\,\sqrt {b\,x^2+a\,x}}{315\,a^2\,x^2}-\frac {20\,b\,\sqrt {b\,x^2+a\,x}}{63\,x^4}-\frac {2\,b^2\,\sqrt {b\,x^2+a\,x}}{105\,a\,x^3}-\frac {2\,a\,\sqrt {b\,x^2+a\,x}}{9\,x^5}-\frac {16\,b^4\,\sqrt {b\,x^2+a\,x}}{315\,a^3\,x} \] Input:
int((a*x + b*x^2)^(3/2)/x^7,x)
Output:
(8*b^3*(a*x + b*x^2)^(1/2))/(315*a^2*x^2) - (20*b*(a*x + b*x^2)^(1/2))/(63 *x^4) - (2*b^2*(a*x + b*x^2)^(1/2))/(105*a*x^3) - (2*a*(a*x + b*x^2)^(1/2) )/(9*x^5) - (16*b^4*(a*x + b*x^2)^(1/2))/(315*a^3*x)
Time = 0.24 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{x^7} \, dx=\frac {-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{4}}{9}-\frac {20 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b x}{63}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{2} x^{2}}{105}+\frac {8 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{3} x^{3}}{315}-\frac {16 \sqrt {x}\, \sqrt {b x +a}\, b^{4} x^{4}}{315}+\frac {16 \sqrt {b}\, b^{4} x^{5}}{315}}{a^{3} x^{5}} \] Input:
int((b*x^2+a*x)^(3/2)/x^7,x)
Output:
(2*( - 35*sqrt(x)*sqrt(a + b*x)*a**4 - 50*sqrt(x)*sqrt(a + b*x)*a**3*b*x - 3*sqrt(x)*sqrt(a + b*x)*a**2*b**2*x**2 + 4*sqrt(x)*sqrt(a + b*x)*a*b**3*x **3 - 8*sqrt(x)*sqrt(a + b*x)*b**4*x**4 + 8*sqrt(b)*b**4*x**5))/(315*a**3* x**5)