Integrand size = 24, antiderivative size = 177 \[ \int \frac {a+b x^2}{(e x)^{11/2} \sqrt {c+d x}} \, dx=-\frac {2 a \sqrt {c+d x}}{9 c e (e x)^{9/2}}+\frac {16 a d \sqrt {c+d x}}{63 c^2 e^2 (e x)^{7/2}}-\frac {2 \left (21 b c^2+16 a d^2\right ) \sqrt {c+d x}}{105 c^3 e^3 (e x)^{5/2}}+\frac {8 d \left (21 b c^2+16 a d^2\right ) \sqrt {c+d x}}{315 c^4 e^4 (e x)^{3/2}}-\frac {16 d^2 \left (21 b c^2+16 a d^2\right ) \sqrt {c+d x}}{315 c^5 e^5 \sqrt {e x}} \] Output:
-2/9*a*(d*x+c)^(1/2)/c/e/(e*x)^(9/2)+16/63*a*d*(d*x+c)^(1/2)/c^2/e^2/(e*x) ^(7/2)-2/105*(16*a*d^2+21*b*c^2)*(d*x+c)^(1/2)/c^3/e^3/(e*x)^(5/2)+8/315*d *(16*a*d^2+21*b*c^2)*(d*x+c)^(1/2)/c^4/e^4/(e*x)^(3/2)-16/315*d^2*(16*a*d^ 2+21*b*c^2)*(d*x+c)^(1/2)/c^5/e^5/(e*x)^(1/2)
Time = 0.21 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.57 \[ \int \frac {a+b x^2}{(e x)^{11/2} \sqrt {c+d x}} \, dx=-\frac {2 x \sqrt {c+d x} \left (35 a c^4-40 a c^3 d x+63 b c^4 x^2+48 a c^2 d^2 x^2-84 b c^3 d x^3-64 a c d^3 x^3+168 b c^2 d^2 x^4+128 a d^4 x^4\right )}{315 c^5 (e x)^{11/2}} \] Input:
Integrate[(a + b*x^2)/((e*x)^(11/2)*Sqrt[c + d*x]),x]
Output:
(-2*x*Sqrt[c + d*x]*(35*a*c^4 - 40*a*c^3*d*x + 63*b*c^4*x^2 + 48*a*c^2*d^2 *x^2 - 84*b*c^3*d*x^3 - 64*a*c*d^3*x^3 + 168*b*c^2*d^2*x^4 + 128*a*d^4*x^4 ))/(315*c^5*(e*x)^(11/2))
Time = 0.40 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {520, 27, 87, 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.
\(\displaystyle \int \frac {a+b x^2}{(e x)^{11/2} \sqrt {c+d x}} \, dx\) |
\(\Big \downarrow \) 520 |
\(\displaystyle -\frac {2 \int \frac {8 a d-9 b c x}{2 (e x)^{9/2} \sqrt {c+d x}}dx}{9 c e}-\frac {2 a \sqrt {c+d x}}{9 c e (e x)^{9/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {8 a d-9 b c x}{(e x)^{9/2} \sqrt {c+d x}}dx}{9 c e}-\frac {2 a \sqrt {c+d x}}{9 c e (e x)^{9/2}}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {-\frac {3 \left (16 a d^2+21 b c^2\right ) \int \frac {1}{(e x)^{7/2} \sqrt {c+d x}}dx}{7 c e}-\frac {16 a d \sqrt {c+d x}}{7 c e (e x)^{7/2}}}{9 c e}-\frac {2 a \sqrt {c+d x}}{9 c e (e x)^{9/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {-\frac {3 \left (16 a d^2+21 b c^2\right ) \left (-\frac {4 d \int \frac {1}{(e x)^{5/2} \sqrt {c+d x}}dx}{5 c e}-\frac {2 \sqrt {c+d x}}{5 c e (e x)^{5/2}}\right )}{7 c e}-\frac {16 a d \sqrt {c+d x}}{7 c e (e x)^{7/2}}}{9 c e}-\frac {2 a \sqrt {c+d x}}{9 c e (e x)^{9/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {-\frac {3 \left (16 a d^2+21 b c^2\right ) \left (-\frac {4 d \left (-\frac {2 d \int \frac {1}{(e x)^{3/2} \sqrt {c+d x}}dx}{3 c e}-\frac {2 \sqrt {c+d x}}{3 c e (e x)^{3/2}}\right )}{5 c e}-\frac {2 \sqrt {c+d x}}{5 c e (e x)^{5/2}}\right )}{7 c e}-\frac {16 a d \sqrt {c+d x}}{7 c e (e x)^{7/2}}}{9 c e}-\frac {2 a \sqrt {c+d x}}{9 c e (e x)^{9/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {-\frac {3 \left (16 a d^2+21 b c^2\right ) \left (-\frac {4 d \left (\frac {4 d \sqrt {c+d x}}{3 c^2 e^2 \sqrt {e x}}-\frac {2 \sqrt {c+d x}}{3 c e (e x)^{3/2}}\right )}{5 c e}-\frac {2 \sqrt {c+d x}}{5 c e (e x)^{5/2}}\right )}{7 c e}-\frac {16 a d \sqrt {c+d x}}{7 c e (e x)^{7/2}}}{9 c e}-\frac {2 a \sqrt {c+d x}}{9 c e (e x)^{9/2}}\) |
Input:
Int[(a + b*x^2)/((e*x)^(11/2)*Sqrt[c + d*x]),x]
Output:
(-2*a*Sqrt[c + d*x])/(9*c*e*(e*x)^(9/2)) - ((-16*a*d*Sqrt[c + d*x])/(7*c*e *(e*x)^(7/2)) - (3*(21*b*c^2 + 16*a*d^2)*((-2*Sqrt[c + d*x])/(5*c*e*(e*x)^ (5/2)) - (4*d*((-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))/(9*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 = 96, normalized size of antiderivative = 0.54
method | result | size |
gosper | \(-\frac {2 x \sqrt {d x +c}\, \left (128 a \,d^{4} x^{4}+168 b \,c^{2} d^{2} x^{4}-64 a c \,d^{3} x^{3}-84 b \,c^{3} d \,x^{3}+48 a \,c^{2} d^{2} x^{2}+63 b \,c^{4} x^{2}-40 a \,c^{3} d x +35 a \,c^{4}\right )}{315 c^{5} \left (e x \right )^{\frac {11}{2}}}\) | \(96\) |
orering | \(-\frac {2 x \sqrt {d x +c}\, \left (128 a \,d^{4} x^{4}+168 b \,c^{2} d^{2} x^{4}-64 a c \,d^{3} x^{3}-84 b \,c^{3} d \,x^{3}+48 a \,c^{2} d^{2} x^{2}+63 b \,c^{4} x^{2}-40 a \,c^{3} d x +35 a \,c^{4}\right )}{315 c^{5} \left (e x \right )^{\frac {11}{2}}}\) | \(96\) |
default | \(-\frac {2 \sqrt {d x +c}\, \left (128 a \,d^{4} x^{4}+168 b \,c^{2} d^{2} x^{4}-64 a c \,d^{3} x^{3}-84 b \,c^{3} d \,x^{3}+48 a \,c^{2} d^{2} x^{2}+63 b \,c^{4} x^{2}-40 a \,c^{3} d x +35 a \,c^{4}\right )}{315 x^{4} c^{5} e^{5} \sqrt {e x}}\) | \(101\) |
risch | \(-\frac {2 \sqrt {d x +c}\, \left (128 a \,d^{4} x^{4}+168 b \,c^{2} d^{2} x^{4}-64 a c \,d^{3} x^{3}-84 b \,c^{3} d \,x^{3}+48 a \,c^{2} d^{2} x^{2}+63 b \,c^{4} x^{2}-40 a \,c^{3} d x +35 a \,c^{4}\right )}{315 x^{4} c^{5} e^{5} \sqrt {e x}}\) | \(101\) |
Input:
int((b*x^2+a)/(e*x)^(11/2)/(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/315*x*(d*x+c)^(1/2)*(128*a*d^4*x^4+168*b*c^2*d^2*x^4-64*a*c*d^3*x^3-84* b*c^3*d*x^3+48*a*c^2*d^2*x^2+63*b*c^4*x^2-40*a*c^3*d*x+35*a*c^4)/c^5/(e*x) ^(11/2)
Time = 0.09 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.56 \[ \int \frac {a+b x^2}{(e x)^{11/2} \sqrt {c+d x}} \, dx=\frac {2 \, {\left (40 \, a c^{3} d x - 35 \, a c^{4} - 8 \, {\left (21 \, b c^{2} d^{2} + 16 \, a d^{4}\right )} x^{4} + 4 \, {\left (21 \, b c^{3} d + 16 \, a c d^{3}\right )} x^{3} - 3 \, {\left (21 \, b c^{4} + 16 \, a c^{2} d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {e x}}{315 \, c^{5} e^{6} x^{5}} \] Input:
integrate((b*x^2+a)/(e*x)^(11/2)/(d*x+c)^(1/2),x, algorithm="fricas")
Output:
2/315*(40*a*c^3*d*x - 35*a*c^4 - 8*(21*b*c^2*d^2 + 16*a*d^4)*x^4 + 4*(21*b *c^3*d + 16*a*c*d^3)*x^3 - 3*(21*b*c^4 + 16*a*c^2*d^2)*x^2)*sqrt(d*x + c)* sqrt(e*x)/(c^5*e^6*x^5)
Leaf count of result is larger than twice the leaf count of optimal. 1357 vs. \(2 (173) = 346\).
Time = 148.98 (sec) , antiderivative size = 1357, normalized size of antiderivative = 7.67 \[ \int \frac {a+b x^2}{(e x)^{11/2} \sqrt {c+d x}} \, dx=\text {Too large to display} \] Input:
integrate((b*x**2+a)/(e*x)**(11/2)/(d*x+c)**(1/2),x)
Output:
-70*a*c**8*d**(33/2)*sqrt(c/(d*x) + 1)/(315*c**9*d**16*e**(11/2)*x**4 + 12 60*c**8*d**17*e**(11/2)*x**5 + 1890*c**7*d**18*e**(11/2)*x**6 + 1260*c**6* d**19*e**(11/2)*x**7 + 315*c**5*d**20*e**(11/2)*x**8) - 200*a*c**7*d**(35/ 2)*x*sqrt(c/(d*x) + 1)/(315*c**9*d**16*e**(11/2)*x**4 + 1260*c**8*d**17*e* *(11/2)*x**5 + 1890*c**7*d**18*e**(11/2)*x**6 + 1260*c**6*d**19*e**(11/2)* x**7 + 315*c**5*d**20*e**(11/2)*x**8) - 196*a*c**6*d**(37/2)*x**2*sqrt(c/( d*x) + 1)/(315*c**9*d**16*e**(11/2)*x**4 + 1260*c**8*d**17*e**(11/2)*x**5 + 1890*c**7*d**18*e**(11/2)*x**6 + 1260*c**6*d**19*e**(11/2)*x**7 + 315*c* *5*d**20*e**(11/2)*x**8) - 56*a*c**5*d**(39/2)*x**3*sqrt(c/(d*x) + 1)/(315 *c**9*d**16*e**(11/2)*x**4 + 1260*c**8*d**17*e**(11/2)*x**5 + 1890*c**7*d* *18*e**(11/2)*x**6 + 1260*c**6*d**19*e**(11/2)*x**7 + 315*c**5*d**20*e**(1 1/2)*x**8) - 70*a*c**4*d**(41/2)*x**4*sqrt(c/(d*x) + 1)/(315*c**9*d**16*e* *(11/2)*x**4 + 1260*c**8*d**17*e**(11/2)*x**5 + 1890*c**7*d**18*e**(11/2)* x**6 + 1260*c**6*d**19*e**(11/2)*x**7 + 315*c**5*d**20*e**(11/2)*x**8) - 5 60*a*c**3*d**(43/2)*x**5*sqrt(c/(d*x) + 1)/(315*c**9*d**16*e**(11/2)*x**4 + 1260*c**8*d**17*e**(11/2)*x**5 + 1890*c**7*d**18*e**(11/2)*x**6 + 1260*c **6*d**19*e**(11/2)*x**7 + 315*c**5*d**20*e**(11/2)*x**8) - 1120*a*c**2*d* *(45/2)*x**6*sqrt(c/(d*x) + 1)/(315*c**9*d**16*e**(11/2)*x**4 + 1260*c**8* d**17*e**(11/2)*x**5 + 1890*c**7*d**18*e**(11/2)*x**6 + 1260*c**6*d**19*e* *(11/2)*x**7 + 315*c**5*d**20*e**(11/2)*x**8) - 896*a*c*d**(47/2)*x**7*...
Exception generated. \[ \int \frac {a+b x^2}{(e x)^{11/2} \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x^2+a)/(e*x)^(11/2)/(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 = 204, normalized size of antiderivative = 1.15 \[ \int \frac {a+b x^2}{(e x)^{11/2} \sqrt {c+d x}} \, dx=-\frac {2 \, {\left ({\left ({\left (d x + c\right )} {\left (4 \, {\left (d x + c\right )} {\left (\frac {2 \, {\left (21 \, b c^{2} d^{7} e^{4} + 16 \, a d^{9} e^{4}\right )} {\left (d x + c\right )}}{c^{5}} - \frac {9 \, {\left (21 \, b c^{3} d^{7} e^{4} + 16 \, a c d^{9} e^{4}\right )}}{c^{5}}\right )} + \frac {63 \, {\left (21 \, b c^{4} d^{7} e^{4} + 16 \, a c^{2} d^{9} e^{4}\right )}}{c^{5}}\right )} - \frac {210 \, {\left (5 \, b c^{5} d^{7} e^{4} + 4 \, a c^{3} d^{9} e^{4}\right )}}{c^{5}}\right )} {\left (d x + c\right )} + \frac {315 \, {\left (b c^{6} d^{7} e^{4} + a c^{4} d^{9} e^{4}\right )}}{c^{5}}\right )} \sqrt {d x + c} d}{315 \, {\left ({\left (d x + c\right )} d e - c d e\right )}^{\frac {9}{2}} e^{5} {\left | d \right |}} \] Input:
integrate((b*x^2+a)/(e*x)^(11/2)/(d*x+c)^(1/2),x, algorithm="giac")
Output:
-2/315*(((d*x + c)*(4*(d*x + c)*(2*(21*b*c^2*d^7*e^4 + 16*a*d^9*e^4)*(d*x + c)/c^5 - 9*(21*b*c^3*d^7*e^4 + 16*a*c*d^9*e^4)/c^5) + 63*(21*b*c^4*d^7*e ^4 + 16*a*c^2*d^9*e^4)/c^5) - 210*(5*b*c^5*d^7*e^4 + 4*a*c^3*d^9*e^4)/c^5) *(d*x + c) + 315*(b*c^6*d^7*e^4 + a*c^4*d^9*e^4)/c^5)*sqrt(d*x + c)*d/(((d *x + c)*d*e - c*d*e)^(9/2)*e^5*abs(d))
Time = 8.83 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.67 \[ \int \frac {a+b x^2}{(e x)^{11/2} \sqrt {c+d x}} \, dx=-\frac {\sqrt {c+d\,x}\,\left (\frac {2\,a}{9\,c\,e^5}+\frac {x^2\,\left (126\,b\,c^4+96\,a\,c^2\,d^2\right )}{315\,c^5\,e^5}+\frac {x^4\,\left (336\,b\,c^2\,d^2+256\,a\,d^4\right )}{315\,c^5\,e^5}-\frac {x^3\,\left (168\,b\,c^3\,d+128\,a\,c\,d^3\right )}{315\,c^5\,e^5}-\frac {16\,a\,d\,x}{63\,c^2\,e^5}\right )}{x^4\,\sqrt {e\,x}} \] Input:
int((a + b*x^2)/((e*x)^(11/2)*(c + d*x)^(1/2)),x)
Output:
-((c + d*x)^(1/2)*((2*a)/(9*c*e^5) + (x^2*(126*b*c^4 + 96*a*c^2*d^2))/(315 *c^5*e^5) + (x^4*(256*a*d^4 + 336*b*c^2*d^2))/(315*c^5*e^5) - (x^3*(128*a* c*d^3 + 168*b*c^3*d))/(315*c^5*e^5) - (16*a*d*x)/(63*c^2*e^5)))/(x^4*(e*x) ^(1/2))
Time = 0.21 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.01 \[ \int \frac {a+b x^2}{(e x)^{11/2} \sqrt {c+d x}} \, dx=\frac {2 \sqrt {e}\, \left (-35 \sqrt {x}\, \sqrt {d x +c}\, a \,c^{4}+40 \sqrt {x}\, \sqrt {d x +c}\, a \,c^{3} d x -48 \sqrt {x}\, \sqrt {d x +c}\, a \,c^{2} d^{2} x^{2}+64 \sqrt {x}\, \sqrt {d x +c}\, a c \,d^{3} x^{3}-128 \sqrt {x}\, \sqrt {d x +c}\, a \,d^{4} x^{4}-63 \sqrt {x}\, \sqrt {d x +c}\, b \,c^{4} x^{2}+84 \sqrt {x}\, \sqrt {d x +c}\, b \,c^{3} d \,x^{3}-168 \sqrt {x}\, \sqrt {d x +c}\, b \,c^{2} d^{2} x^{4}+128 \sqrt {d}\, a \,d^{4} x^{5}+168 \sqrt {d}\, b \,c^{2} d^{2} x^{5}\right )}{315 c^{5} e^{6} x^{5}} \] Input:
int((b*x^2+a)/(e*x)^(11/2)/(d*x+c)^(1/2),x)
Output:
(2*sqrt(e)*( - 35*sqrt(x)*sqrt(c + d*x)*a*c**4 + 40*sqrt(x)*sqrt(c + d*x)* a*c**3*d*x - 48*sqrt(x)*sqrt(c + d*x)*a*c**2*d**2*x**2 + 64*sqrt(x)*sqrt(c + d*x)*a*c*d**3*x**3 - 128*sqrt(x)*sqrt(c + d*x)*a*d**4*x**4 - 63*sqrt(x) *sqrt(c + d*x)*b*c**4*x**2 + 84*sqrt(x)*sqrt(c + d*x)*b*c**3*d*x**3 - 168* sqrt(x)*sqrt(c + d*x)*b*c**2*d**2*x**4 + 128*sqrt(d)*a*d**4*x**5 + 168*sqr t(d)*b*c**2*d**2*x**5))/(315*c**5*e**6*x**5)