Integrand size = 24, antiderivative size = 135 \[ \int \frac {a+b x^2}{(e x)^{9/2} \sqrt {c+d x}} \, dx=-\frac {2 a \sqrt {c+d x}}{7 c e (e x)^{7/2}}+\frac {12 a d \sqrt {c+d x}}{35 c^2 e^2 (e x)^{5/2}}-\frac {2 \left (35 b c^2+24 a d^2\right ) \sqrt {c+d x}}{105 c^3 e^3 (e x)^{3/2}}+\frac {4 d \left (35 b c^2+24 a d^2\right ) \sqrt {c+d x}}{105 c^4 e^4 \sqrt {e x}} \] Output:
-2/7*a*(d*x+c)^(1/2)/c/e/(e*x)^(7/2)+12/35*a*d*(d*x+c)^(1/2)/c^2/e^2/(e*x) ^(5/2)-2/105*(24*a*d^2+35*b*c^2)*(d*x+c)^(1/2)/c^3/e^3/(e*x)^(3/2)+4/105*d *(24*a*d^2+35*b*c^2)*(d*x+c)^(1/2)/c^4/e^4/(e*x)^(1/2)
Time = 0.15 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.54 \[ \int \frac {a+b x^2}{(e x)^{9/2} \sqrt {c+d x}} \, dx=-\frac {2 x \sqrt {c+d x} \left (35 b c^2 x^2 (c-2 d x)+3 a \left (5 c^3-6 c^2 d x+8 c d^2 x^2-16 d^3 x^3\right )\right )}{105 c^4 (e x)^{9/2}} \] Input:
Integrate[(a + b*x^2)/((e*x)^(9/2)*Sqrt[c + d*x]),x]
Output:
(-2*x*Sqrt[c + d*x]*(35*b*c^2*x^2*(c - 2*d*x) + 3*a*(5*c^3 - 6*c^2*d*x + 8 *c*d^2*x^2 - 16*d^3*x^3)))/(105*c^4*(e*x)^(9/2))
Time = 0.39 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {520, 27, 87, 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.
\(\displaystyle \int \frac {a+b x^2}{(e x)^{9/2} \sqrt {c+d x}} \, dx\) |
\(\Big \downarrow \) 520 |
\(\displaystyle -\frac {2 \int \frac {6 a d-7 b c x}{2 (e x)^{7/2} \sqrt {c+d x}}dx}{7 c e}-\frac {2 a \sqrt {c+d x}}{7 c e (e x)^{7/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {6 a d-7 b c x}{(e x)^{7/2} \sqrt {c+d x}}dx}{7 c e}-\frac {2 a \sqrt {c+d x}}{7 c e (e x)^{7/2}}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {-\frac {\left (24 a d^2+35 b c^2\right ) \int \frac {1}{(e x)^{5/2} \sqrt {c+d x}}dx}{5 c e}-\frac {12 a d \sqrt {c+d x}}{5 c e (e x)^{5/2}}}{7 c e}-\frac {2 a \sqrt {c+d x}}{7 c e (e x)^{7/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {-\frac {\left (24 a d^2+35 b c^2\right ) \left (-\frac {2 d \int \frac {1}{(e x)^{3/2} \sqrt {c+d x}}dx}{3 c e}-\frac {2 \sqrt {c+d x}}{3 c e (e x)^{3/2}}\right )}{5 c e}-\frac {12 a d \sqrt {c+d x}}{5 c e (e x)^{5/2}}}{7 c e}-\frac {2 a \sqrt {c+d x}}{7 c e (e x)^{7/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {-\frac {\left (24 a d^2+35 b c^2\right ) \left (\frac {4 d \sqrt {c+d x}}{3 c^2 e^2 \sqrt {e x}}-\frac {2 \sqrt {c+d x}}{3 c e (e x)^{3/2}}\right )}{5 c e}-\frac {12 a d \sqrt {c+d x}}{5 c e (e x)^{5/2}}}{7 c e}-\frac {2 a \sqrt {c+d x}}{7 c e (e x)^{7/2}}\) |
Input:
Int[(a + b*x^2)/((e*x)^(9/2)*Sqrt[c + d*x]),x]
Output:
(-2*a*Sqrt[c + d*x])/(7*c*e*(e*x)^(7/2)) - ((-12*a*d*Sqrt[c + d*x])/(5*c*e *(e*x)^(5/2)) - ((35*b*c^2 + 24*a*d^2)*((-2*Sqrt[c + d*x])/(3*c*e*(e*x)^(3 /2)) + (4*d*Sqrt[c + d*x])/(3*c^2*e^2*Sqrt[e*x])))/(5*c*e))/(7*c*e)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(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] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2)^p, e*x, x], R = Pol ynomialRemainder[(a + b*x^2)^p, e*x, x]}, Simp[R*(e*x)^(m + 1)*((c + d*x)^( n + 1)/((m + 1)*(e*c))), x] + Simp[1/((m + 1)*(e*c)) Int[(e*x)^(m + 1)*(c + d*x)^n*ExpandToSum[(m + 1)*(e*c)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr eeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[m, -1] && !IntegerQ[n]
Time = 0.24 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.53
method | result | size |
gosper | \(-\frac {2 x \sqrt {d x +c}\, \left (-48 a \,x^{3} d^{3}-70 b \,c^{2} d \,x^{3}+24 a \,d^{2} x^{2} c +35 b \,c^{3} x^{2}-18 a d x \,c^{2}+15 c^{3} a \right )}{105 c^{4} \left (e x \right )^{\frac {9}{2}}}\) | \(72\) |
orering | \(-\frac {2 x \sqrt {d x +c}\, \left (-48 a \,x^{3} d^{3}-70 b \,c^{2} d \,x^{3}+24 a \,d^{2} x^{2} c +35 b \,c^{3} x^{2}-18 a d x \,c^{2}+15 c^{3} a \right )}{105 c^{4} \left (e x \right )^{\frac {9}{2}}}\) | \(72\) |
default | \(-\frac {2 \sqrt {d x +c}\, \left (-48 a \,x^{3} d^{3}-70 b \,c^{2} d \,x^{3}+24 a \,d^{2} x^{2} c +35 b \,c^{3} x^{2}-18 a d x \,c^{2}+15 c^{3} a \right )}{105 x^{3} c^{4} e^{4} \sqrt {e x}}\) | \(77\) |
risch | \(-\frac {2 \sqrt {d x +c}\, \left (-48 a \,x^{3} d^{3}-70 b \,c^{2} d \,x^{3}+24 a \,d^{2} x^{2} c +35 b \,c^{3} x^{2}-18 a d x \,c^{2}+15 c^{3} a \right )}{105 x^{3} c^{4} e^{4} \sqrt {e x}}\) | \(77\) |
Input:
int((b*x^2+a)/(e*x)^(9/2)/(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/105*x*(d*x+c)^(1/2)*(-48*a*d^3*x^3-70*b*c^2*d*x^3+24*a*c*d^2*x^2+35*b*c ^3*x^2-18*a*c^2*d*x+15*a*c^3)/c^4/(e*x)^(9/2)
Time = 0.09 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.56 \[ \int \frac {a+b x^2}{(e x)^{9/2} \sqrt {c+d x}} \, dx=\frac {2 \, {\left (18 \, a c^{2} d x - 15 \, a c^{3} + 2 \, {\left (35 \, b c^{2} d + 24 \, a d^{3}\right )} x^{3} - {\left (35 \, b c^{3} + 24 \, a c d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {e x}}{105 \, c^{4} e^{5} x^{4}} \] Input:
integrate((b*x^2+a)/(e*x)^(9/2)/(d*x+c)^(1/2),x, algorithm="fricas")
Output:
2/105*(18*a*c^2*d*x - 15*a*c^3 + 2*(35*b*c^2*d + 24*a*d^3)*x^3 - (35*b*c^3 + 24*a*c*d^2)*x^2)*sqrt(d*x + c)*sqrt(e*x)/(c^4*e^5*x^4)
Leaf count of result is larger than twice the leaf count of optimal. 700 vs. \(2 (131) = 262\).
Time = 48.23 (sec) , antiderivative size = 700, normalized size of antiderivative = 5.19 \[ \int \frac {a+b x^2}{(e x)^{9/2} \sqrt {c+d x}} \, dx=- \frac {10 a c^{6} d^{\frac {19}{2}} \sqrt {\frac {c}{d x} + 1}}{35 c^{7} d^{9} e^{\frac {9}{2}} x^{3} + 105 c^{6} d^{10} e^{\frac {9}{2}} x^{4} + 105 c^{5} d^{11} e^{\frac {9}{2}} x^{5} + 35 c^{4} d^{12} e^{\frac {9}{2}} x^{6}} - \frac {18 a c^{5} d^{\frac {21}{2}} x \sqrt {\frac {c}{d x} + 1}}{35 c^{7} d^{9} e^{\frac {9}{2}} x^{3} + 105 c^{6} d^{10} e^{\frac {9}{2}} x^{4} + 105 c^{5} d^{11} e^{\frac {9}{2}} x^{5} + 35 c^{4} d^{12} e^{\frac {9}{2}} x^{6}} - \frac {10 a c^{4} d^{\frac {23}{2}} x^{2} \sqrt {\frac {c}{d x} + 1}}{35 c^{7} d^{9} e^{\frac {9}{2}} x^{3} + 105 c^{6} d^{10} e^{\frac {9}{2}} x^{4} + 105 c^{5} d^{11} e^{\frac {9}{2}} x^{5} + 35 c^{4} d^{12} e^{\frac {9}{2}} x^{6}} + \frac {10 a c^{3} d^{\frac {25}{2}} x^{3} \sqrt {\frac {c}{d x} + 1}}{35 c^{7} d^{9} e^{\frac {9}{2}} x^{3} + 105 c^{6} d^{10} e^{\frac {9}{2}} x^{4} + 105 c^{5} d^{11} e^{\frac {9}{2}} x^{5} + 35 c^{4} d^{12} e^{\frac {9}{2}} x^{6}} + \frac {60 a c^{2} d^{\frac {27}{2}} x^{4} \sqrt {\frac {c}{d x} + 1}}{35 c^{7} d^{9} e^{\frac {9}{2}} x^{3} + 105 c^{6} d^{10} e^{\frac {9}{2}} x^{4} + 105 c^{5} d^{11} e^{\frac {9}{2}} x^{5} + 35 c^{4} d^{12} e^{\frac {9}{2}} x^{6}} + \frac {80 a c d^{\frac {29}{2}} x^{5} \sqrt {\frac {c}{d x} + 1}}{35 c^{7} d^{9} e^{\frac {9}{2}} x^{3} + 105 c^{6} d^{10} e^{\frac {9}{2}} x^{4} + 105 c^{5} d^{11} e^{\frac {9}{2}} x^{5} + 35 c^{4} d^{12} e^{\frac {9}{2}} x^{6}} + \frac {32 a d^{\frac {31}{2}} x^{6} \sqrt {\frac {c}{d x} + 1}}{35 c^{7} d^{9} e^{\frac {9}{2}} x^{3} + 105 c^{6} d^{10} e^{\frac {9}{2}} x^{4} + 105 c^{5} d^{11} e^{\frac {9}{2}} x^{5} + 35 c^{4} d^{12} e^{\frac {9}{2}} x^{6}} - \frac {2 b \sqrt {d} \sqrt {\frac {c}{d x} + 1}}{3 c e^{\frac {9}{2}} x} + \frac {4 b d^{\frac {3}{2}} \sqrt {\frac {c}{d x} + 1}}{3 c^{2} e^{\frac {9}{2}}} \] Input:
integrate((b*x**2+a)/(e*x)**(9/2)/(d*x+c)**(1/2),x)
Output:
-10*a*c**6*d**(19/2)*sqrt(c/(d*x) + 1)/(35*c**7*d**9*e**(9/2)*x**3 + 105*c **6*d**10*e**(9/2)*x**4 + 105*c**5*d**11*e**(9/2)*x**5 + 35*c**4*d**12*e** (9/2)*x**6) - 18*a*c**5*d**(21/2)*x*sqrt(c/(d*x) + 1)/(35*c**7*d**9*e**(9/ 2)*x**3 + 105*c**6*d**10*e**(9/2)*x**4 + 105*c**5*d**11*e**(9/2)*x**5 + 35 *c**4*d**12*e**(9/2)*x**6) - 10*a*c**4*d**(23/2)*x**2*sqrt(c/(d*x) + 1)/(3 5*c**7*d**9*e**(9/2)*x**3 + 105*c**6*d**10*e**(9/2)*x**4 + 105*c**5*d**11* e**(9/2)*x**5 + 35*c**4*d**12*e**(9/2)*x**6) + 10*a*c**3*d**(25/2)*x**3*sq rt(c/(d*x) + 1)/(35*c**7*d**9*e**(9/2)*x**3 + 105*c**6*d**10*e**(9/2)*x**4 + 105*c**5*d**11*e**(9/2)*x**5 + 35*c**4*d**12*e**(9/2)*x**6) + 60*a*c**2 *d**(27/2)*x**4*sqrt(c/(d*x) + 1)/(35*c**7*d**9*e**(9/2)*x**3 + 105*c**6*d **10*e**(9/2)*x**4 + 105*c**5*d**11*e**(9/2)*x**5 + 35*c**4*d**12*e**(9/2) *x**6) + 80*a*c*d**(29/2)*x**5*sqrt(c/(d*x) + 1)/(35*c**7*d**9*e**(9/2)*x* *3 + 105*c**6*d**10*e**(9/2)*x**4 + 105*c**5*d**11*e**(9/2)*x**5 + 35*c**4 *d**12*e**(9/2)*x**6) + 32*a*d**(31/2)*x**6*sqrt(c/(d*x) + 1)/(35*c**7*d** 9*e**(9/2)*x**3 + 105*c**6*d**10*e**(9/2)*x**4 + 105*c**5*d**11*e**(9/2)*x **5 + 35*c**4*d**12*e**(9/2)*x**6) - 2*b*sqrt(d)*sqrt(c/(d*x) + 1)/(3*c*e* *(9/2)*x) + 4*b*d**(3/2)*sqrt(c/(d*x) + 1)/(3*c**2*e**(9/2))
Exception generated. \[ \int \frac {a+b x^2}{(e x)^{9/2} \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x^2+a)/(e*x)^(9/2)/(d*x+c)^(1/2),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.14 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.19 \[ \int \frac {a+b x^2}{(e x)^{9/2} \sqrt {c+d x}} \, dx=\frac {2 \, {\left ({\left (d x + c\right )} {\left ({\left (d x + c\right )} {\left (\frac {2 \, {\left (35 \, b c^{2} d e^{3} + 24 \, a d^{3} e^{3}\right )} {\left (d x + c\right )}}{c^{4}} - \frac {7 \, {\left (35 \, b c^{3} d e^{3} + 24 \, a c d^{3} e^{3}\right )}}{c^{4}}\right )} + \frac {70 \, {\left (4 \, b c^{4} d e^{3} + 3 \, a c^{2} d^{3} e^{3}\right )}}{c^{4}}\right )} - \frac {105 \, {\left (b c^{5} d e^{3} + a c^{3} d^{3} e^{3}\right )}}{c^{4}}\right )} \sqrt {d x + c} d^{5}}{105 \, {\left ({\left (d x + c\right )} d e - c d e\right )}^{\frac {7}{2}} e^{4} {\left | d \right |}} \] Input:
integrate((b*x^2+a)/(e*x)^(9/2)/(d*x+c)^(1/2),x, algorithm="giac")
Output:
2/105*((d*x + c)*((d*x + c)*(2*(35*b*c^2*d*e^3 + 24*a*d^3*e^3)*(d*x + c)/c ^4 - 7*(35*b*c^3*d*e^3 + 24*a*c*d^3*e^3)/c^4) + 70*(4*b*c^4*d*e^3 + 3*a*c^ 2*d^3*e^3)/c^4) - 105*(b*c^5*d*e^3 + a*c^3*d^3*e^3)/c^4)*sqrt(d*x + c)*d^5 /(((d*x + c)*d*e - c*d*e)^(7/2)*e^4*abs(d))
Time = 8.48 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.65 \[ \int \frac {a+b x^2}{(e x)^{9/2} \sqrt {c+d x}} \, dx=-\frac {\sqrt {c+d\,x}\,\left (\frac {2\,a}{7\,c\,e^4}+\frac {x^2\,\left (70\,b\,c^3+48\,a\,c\,d^2\right )}{105\,c^4\,e^4}-\frac {x^3\,\left (140\,b\,c^2\,d+96\,a\,d^3\right )}{105\,c^4\,e^4}-\frac {12\,a\,d\,x}{35\,c^2\,e^4}\right )}{x^3\,\sqrt {e\,x}} \] Input:
int((a + b*x^2)/((e*x)^(9/2)*(c + d*x)^(1/2)),x)
Output:
-((c + d*x)^(1/2)*((2*a)/(7*c*e^4) + (x^2*(70*b*c^3 + 48*a*c*d^2))/(105*c^ 4*e^4) - (x^3*(96*a*d^3 + 140*b*c^2*d))/(105*c^4*e^4) - (12*a*d*x)/(35*c^2 *e^4)))/(x^3*(e*x)^(1/2))
Time = 0.20 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.01 \[ \int \frac {a+b x^2}{(e x)^{9/2} \sqrt {c+d x}} \, dx=\frac {2 \sqrt {e}\, \left (-15 \sqrt {x}\, \sqrt {d x +c}\, a \,c^{3}+18 \sqrt {x}\, \sqrt {d x +c}\, a \,c^{2} d x -24 \sqrt {x}\, \sqrt {d x +c}\, a c \,d^{2} x^{2}+48 \sqrt {x}\, \sqrt {d x +c}\, a \,d^{3} x^{3}-35 \sqrt {x}\, \sqrt {d x +c}\, b \,c^{3} x^{2}+70 \sqrt {x}\, \sqrt {d x +c}\, b \,c^{2} d \,x^{3}-48 \sqrt {d}\, a \,d^{3} x^{4}-70 \sqrt {d}\, b \,c^{2} d \,x^{4}\right )}{105 c^{4} e^{5} x^{4}} \] Input:
int((b*x^2+a)/(e*x)^(9/2)/(d*x+c)^(1/2),x)
Output:
(2*sqrt(e)*( - 15*sqrt(x)*sqrt(c + d*x)*a*c**3 + 18*sqrt(x)*sqrt(c + d*x)* a*c**2*d*x - 24*sqrt(x)*sqrt(c + d*x)*a*c*d**2*x**2 + 48*sqrt(x)*sqrt(c + d*x)*a*d**3*x**3 - 35*sqrt(x)*sqrt(c + d*x)*b*c**3*x**2 + 70*sqrt(x)*sqrt( c + d*x)*b*c**2*d*x**3 - 48*sqrt(d)*a*d**3*x**4 - 70*sqrt(d)*b*c**2*d*x**4 ))/(105*c**4*e**5*x**4)