Integrand size = 24, antiderivative size = 135 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )}{(e x)^{13/2}} \, dx=-\frac {2 a (c+d x)^{5/2}}{11 c e (e x)^{11/2}}+\frac {4 a d (c+d x)^{5/2}}{33 c^2 e^2 (e x)^{9/2}}-\frac {2 \left (33 b c^2+8 a d^2\right ) (c+d x)^{5/2}}{231 c^3 e^3 (e x)^{7/2}}+\frac {4 d \left (33 b c^2+8 a d^2\right ) (c+d x)^{5/2}}{1155 c^4 e^4 (e x)^{5/2}} \] Output:
-2/11*a*(d*x+c)^(5/2)/c/e/(e*x)^(11/2)+4/33*a*d*(d*x+c)^(5/2)/c^2/e^2/(e*x )^(9/2)-2/231*(8*a*d^2+33*b*c^2)*(d*x+c)^(5/2)/c^3/e^3/(e*x)^(7/2)+4/1155* d*(8*a*d^2+33*b*c^2)*(d*x+c)^(5/2)/c^4/e^4/(e*x)^(5/2)
Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.55 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )}{(e x)^{13/2}} \, dx=-\frac {2 x (c+d x)^{5/2} \left (33 b c^2 x^2 (5 c-2 d x)+a \left (105 c^3-70 c^2 d x+40 c d^2 x^2-16 d^3 x^3\right )\right )}{1155 c^4 (e x)^{13/2}} \] Input:
Integrate[((c + d*x)^(3/2)*(a + b*x^2))/(e*x)^(13/2),x]
Output:
(-2*x*(c + d*x)^(5/2)*(33*b*c^2*x^2*(5*c - 2*d*x) + a*(105*c^3 - 70*c^2*d* x + 40*c*d^2*x^2 - 16*d^3*x^3)))/(1155*c^4*(e*x)^(13/2))
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.
\(\displaystyle \int \frac {\left (a+b x^2\right ) (c+d x)^{3/2}}{(e x)^{13/2}} \, dx\) |
\(\Big \downarrow \) 520 |
\(\displaystyle -\frac {2 \int \frac {(6 a d-11 b c x) (c+d x)^{3/2}}{2 (e x)^{11/2}}dx}{11 c e}-\frac {2 a (c+d x)^{5/2}}{11 c e (e x)^{11/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {(6 a d-11 b c x) (c+d x)^{3/2}}{(e x)^{11/2}}dx}{11 c e}-\frac {2 a (c+d x)^{5/2}}{11 c e (e x)^{11/2}}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {-\frac {\left (8 a d^2+33 b c^2\right ) \int \frac {(c+d x)^{3/2}}{(e x)^{9/2}}dx}{3 c e}-\frac {4 a d (c+d x)^{5/2}}{3 c e (e x)^{9/2}}}{11 c e}-\frac {2 a (c+d x)^{5/2}}{11 c e (e x)^{11/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {-\frac {\left (8 a d^2+33 b c^2\right ) \left (-\frac {2 d \int \frac {(c+d x)^{3/2}}{(e x)^{7/2}}dx}{7 c e}-\frac {2 (c+d x)^{5/2}}{7 c e (e x)^{7/2}}\right )}{3 c e}-\frac {4 a d (c+d x)^{5/2}}{3 c e (e x)^{9/2}}}{11 c e}-\frac {2 a (c+d x)^{5/2}}{11 c e (e x)^{11/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {-\frac {\left (8 a d^2+33 b c^2\right ) \left (\frac {4 d (c+d x)^{5/2}}{35 c^2 e^2 (e x)^{5/2}}-\frac {2 (c+d x)^{5/2}}{7 c e (e x)^{7/2}}\right )}{3 c e}-\frac {4 a d (c+d x)^{5/2}}{3 c e (e x)^{9/2}}}{11 c e}-\frac {2 a (c+d x)^{5/2}}{11 c e (e x)^{11/2}}\) |
Input:
Int[((c + d*x)^(3/2)*(a + b*x^2))/(e*x)^(13/2),x]
Output:
(-2*a*(c + d*x)^(5/2))/(11*c*e*(e*x)^(11/2)) - ((-4*a*d*(c + d*x)^(5/2))/( 3*c*e*(e*x)^(9/2)) - ((33*b*c^2 + 8*a*d^2)*((-2*(c + d*x)^(5/2))/(7*c*e*(e *x)^(7/2)) + (4*d*(c + d*x)^(5/2))/(35*c^2*e^2*(e*x)^(5/2))))/(3*c*e))/(11 *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 = 0.25 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.53
method | result | size |
gosper | \(-\frac {2 x \left (d x +c \right )^{\frac {5}{2}} \left (-16 a \,x^{3} d^{3}-66 b \,c^{2} d \,x^{3}+40 a \,d^{2} x^{2} c +165 b \,c^{3} x^{2}-70 a d x \,c^{2}+105 c^{3} a \right )}{1155 c^{4} \left (e x \right )^{\frac {13}{2}}}\) | \(72\) |
orering | \(-\frac {2 x \left (d x +c \right )^{\frac {5}{2}} \left (-16 a \,x^{3} d^{3}-66 b \,c^{2} d \,x^{3}+40 a \,d^{2} x^{2} c +165 b \,c^{3} x^{2}-70 a d x \,c^{2}+105 c^{3} a \right )}{1155 c^{4} \left (e x \right )^{\frac {13}{2}}}\) | \(72\) |
default | \(-\frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (-16 a \,d^{4} x^{4}-66 b \,c^{2} d^{2} x^{4}+24 a c \,d^{3} x^{3}+99 b \,c^{3} d \,x^{3}-30 a \,c^{2} d^{2} x^{2}+165 b \,c^{4} x^{2}+35 a \,c^{3} d x +105 a \,c^{4}\right )}{1155 x^{5} c^{4} e^{6} \sqrt {e x}}\) | \(101\) |
risch | \(-\frac {2 \sqrt {d x +c}\, \left (-16 a \,d^{5} x^{5}-66 b \,c^{2} d^{3} x^{5}+8 a c \,d^{4} x^{4}+33 b \,c^{3} d^{2} x^{4}-6 a \,c^{2} d^{3} x^{3}+264 b \,c^{4} d \,x^{3}+5 a \,c^{3} d^{2} x^{2}+165 b \,c^{5} x^{2}+140 a \,c^{4} d x +105 a \,c^{5}\right )}{1155 e^{6} \sqrt {e x}\, x^{5} c^{4}}\) | \(125\) |
Input:
int((d*x+c)^(3/2)*(b*x^2+a)/(e*x)^(13/2),x,method=_RETURNVERBOSE)
Output:
-2/1155*x*(d*x+c)^(5/2)*(-16*a*d^3*x^3-66*b*c^2*d*x^3+40*a*c*d^2*x^2+165*b *c^3*x^2-70*a*c^2*d*x+105*a*c^3)/c^4/(e*x)^(13/2)
Time = 0.09 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.90 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )}{(e x)^{13/2}} \, dx=-\frac {2 \, {\left (140 \, a c^{4} d x + 105 \, a c^{5} - 2 \, {\left (33 \, b c^{2} d^{3} + 8 \, a d^{5}\right )} x^{5} + {\left (33 \, b c^{3} d^{2} + 8 \, a c d^{4}\right )} x^{4} + 6 \, {\left (44 \, b c^{4} d - a c^{2} d^{3}\right )} x^{3} + 5 \, {\left (33 \, b c^{5} + a c^{3} d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {e x}}{1155 \, c^{4} e^{7} x^{6}} \] Input:
integrate((d*x+c)^(3/2)*(b*x^2+a)/(e*x)^(13/2),x, algorithm="fricas")
Output:
-2/1155*(140*a*c^4*d*x + 105*a*c^5 - 2*(33*b*c^2*d^3 + 8*a*d^5)*x^5 + (33* b*c^3*d^2 + 8*a*c*d^4)*x^4 + 6*(44*b*c^4*d - a*c^2*d^3)*x^3 + 5*(33*b*c^5 + a*c^3*d^2)*x^2)*sqrt(d*x + c)*sqrt(e*x)/(c^4*e^7*x^6)
Timed out. \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )}{(e x)^{13/2}} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**(3/2)*(b*x**2+a)/(e*x)**(13/2),x)
Output:
Timed out
Exception generated. \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )}{(e x)^{13/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)^(3/2)*(b*x^2+a)/(e*x)^(13/2),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 = 0.15 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.27 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )}{(e x)^{13/2}} \, dx=\frac {2 \, {\left ({\left (d x + c\right )} {\left ({\left (d x + c\right )} {\left (\frac {2 \, {\left (33 \, b c^{3} d^{3} e^{5} + 8 \, a c d^{5} e^{5}\right )} {\left (d x + c\right )}}{c^{5}} - \frac {11 \, {\left (33 \, b c^{4} d^{3} e^{5} + 8 \, a c^{2} d^{5} e^{5}\right )}}{c^{5}}\right )} + \frac {66 \, {\left (8 \, b c^{5} d^{3} e^{5} + 3 \, a c^{3} d^{5} e^{5}\right )}}{c^{5}}\right )} - \frac {231 \, {\left (b c^{6} d^{3} e^{5} + a c^{4} d^{5} e^{5}\right )}}{c^{5}}\right )} {\left (d x + c\right )}^{\frac {5}{2}} d^{7}}{1155 \, {\left ({\left (d x + c\right )} d e - c d e\right )}^{\frac {11}{2}} e^{6} {\left | d \right |}} \] Input:
integrate((d*x+c)^(3/2)*(b*x^2+a)/(e*x)^(13/2),x, algorithm="giac")
Output:
2/1155*((d*x + c)*((d*x + c)*(2*(33*b*c^3*d^3*e^5 + 8*a*c*d^5*e^5)*(d*x + c)/c^5 - 11*(33*b*c^4*d^3*e^5 + 8*a*c^2*d^5*e^5)/c^5) + 66*(8*b*c^5*d^3*e^ 5 + 3*a*c^3*d^5*e^5)/c^5) - 231*(b*c^6*d^3*e^5 + a*c^4*d^5*e^5)/c^5)*(d*x + c)^(5/2)*d^7/(((d*x + c)*d*e - c*d*e)^(11/2)*e^6*abs(d))
Time = 9.41 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.06 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )}{(e x)^{13/2}} \, dx=-\frac {\sqrt {c+d\,x}\,\left (\frac {2\,a\,c}{11\,e^6}+\frac {x^4\,\left (66\,b\,c^3\,d^2+16\,a\,c\,d^4\right )}{1155\,c^4\,e^6}-\frac {x^3\,\left (12\,a\,c^2\,d^3-528\,b\,c^4\,d\right )}{1155\,c^4\,e^6}+\frac {8\,a\,d\,x}{33\,e^6}+\frac {x^2\,\left (330\,b\,c^5+10\,a\,c^3\,d^2\right )}{1155\,c^4\,e^6}-\frac {x^5\,\left (132\,b\,c^2\,d^3+32\,a\,d^5\right )}{1155\,c^4\,e^6}\right )}{x^5\,\sqrt {e\,x}} \] Input:
int(((a + b*x^2)*(c + d*x)^(3/2))/(e*x)^(13/2),x)
Output:
-((c + d*x)^(1/2)*((2*a*c)/(11*e^6) + (x^4*(66*b*c^3*d^2 + 16*a*c*d^4))/(1 155*c^4*e^6) - (x^3*(12*a*c^2*d^3 - 528*b*c^4*d))/(1155*c^4*e^6) + (8*a*d* x)/(33*e^6) + (x^2*(330*b*c^5 + 10*a*c^3*d^2))/(1155*c^4*e^6) - (x^5*(32*a *d^5 + 132*b*c^2*d^3))/(1155*c^4*e^6)))/(x^5*(e*x)^(1/2))
Time = 0.23 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.62 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )}{(e x)^{13/2}} \, dx=\frac {2 \sqrt {e}\, \left (-105 \sqrt {x}\, \sqrt {d x +c}\, a \,c^{5}-140 \sqrt {x}\, \sqrt {d x +c}\, a \,c^{4} d x -5 \sqrt {x}\, \sqrt {d x +c}\, a \,c^{3} d^{2} x^{2}+6 \sqrt {x}\, \sqrt {d x +c}\, a \,c^{2} d^{3} x^{3}-8 \sqrt {x}\, \sqrt {d x +c}\, a c \,d^{4} x^{4}+16 \sqrt {x}\, \sqrt {d x +c}\, a \,d^{5} x^{5}-165 \sqrt {x}\, \sqrt {d x +c}\, b \,c^{5} x^{2}-264 \sqrt {x}\, \sqrt {d x +c}\, b \,c^{4} d \,x^{3}-33 \sqrt {x}\, \sqrt {d x +c}\, b \,c^{3} d^{2} x^{4}+66 \sqrt {x}\, \sqrt {d x +c}\, b \,c^{2} d^{3} x^{5}-16 \sqrt {d}\, a \,d^{5} x^{6}-66 \sqrt {d}\, b \,c^{2} d^{3} x^{6}\right )}{1155 c^{4} e^{7} x^{6}} \] Input:
int((d*x+c)^(3/2)*(b*x^2+a)/(e*x)^(13/2),x)
Output:
(2*sqrt(e)*( - 105*sqrt(x)*sqrt(c + d*x)*a*c**5 - 140*sqrt(x)*sqrt(c + d*x )*a*c**4*d*x - 5*sqrt(x)*sqrt(c + d*x)*a*c**3*d**2*x**2 + 6*sqrt(x)*sqrt(c + d*x)*a*c**2*d**3*x**3 - 8*sqrt(x)*sqrt(c + d*x)*a*c*d**4*x**4 + 16*sqrt (x)*sqrt(c + d*x)*a*d**5*x**5 - 165*sqrt(x)*sqrt(c + d*x)*b*c**5*x**2 - 26 4*sqrt(x)*sqrt(c + d*x)*b*c**4*d*x**3 - 33*sqrt(x)*sqrt(c + d*x)*b*c**3*d* *2*x**4 + 66*sqrt(x)*sqrt(c + d*x)*b*c**2*d**3*x**5 - 16*sqrt(d)*a*d**5*x* *6 - 66*sqrt(d)*b*c**2*d**3*x**6))/(1155*c**4*e**7*x**6)