Integrand size = 24, antiderivative size = 95 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{11/2}} \, dx=-\frac {2 (B d-A e) (a+b x)^{7/2}}{9 e (b d-a e) (d+e x)^{9/2}}+\frac {2 (7 b B d+2 A b e-9 a B e) (a+b x)^{7/2}}{63 e (b d-a e)^2 (d+e x)^{7/2}} \] Output:
-2/9*(-A*e+B*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)/(e*x+d)^(9/2)+2/63*(2*A*b*e-9*B *a*e+7*B*b*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)^2/(e*x+d)^(7/2)
Time = 0.14 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.69 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{11/2}} \, dx=\frac {2 (a+b x)^{7/2} (B (-2 a d+7 b d x-9 a e x)+A (9 b d-7 a e+2 b e x))}{63 (b d-a e)^2 (d+e x)^{9/2}} \] Input:
Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(11/2),x]
Output:
(2*(a + b*x)^(7/2)*(B*(-2*a*d + 7*b*d*x - 9*a*e*x) + A*(9*b*d - 7*a*e + 2* b*e*x)))/(63*(b*d - a*e)^2*(d + e*x)^(9/2))
Time = 0.20 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {87, 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.
\(\displaystyle \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{11/2}} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {(-9 a B e+2 A b e+7 b B d) \int \frac {(a+b x)^{5/2}}{(d+e x)^{9/2}}dx}{9 e (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {2 (a+b x)^{7/2} (-9 a B e+2 A b e+7 b B d)}{63 e (d+e x)^{7/2} (b d-a e)^2}-\frac {2 (a+b x)^{7/2} (B d-A e)}{9 e (d+e x)^{9/2} (b d-a e)}\) |
Input:
Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(11/2),x]
Output:
(-2*(B*d - A*e)*(a + b*x)^(7/2))/(9*e*(b*d - a*e)*(d + e*x)^(9/2)) + (2*(7 *b*B*d + 2*A*b*e - 9*a*B*e)*(a + b*x)^(7/2))/(63*e*(b*d - a*e)^2*(d + e*x) ^(7/2))
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_))*((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]))))
Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.78
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-2 A b e x +9 B a e x -7 B b d x +7 A a e -9 A b d +2 B a d \right )}{63 \left (e x +d \right )^{\frac {9}{2}} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}\) | \(74\) |
orering | \(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-2 A b e x +9 B a e x -7 B b d x +7 A a e -9 A b d +2 B a d \right )}{63 \left (e x +d \right )^{\frac {9}{2}} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}\) | \(74\) |
default | \(-\frac {2 \left (-2 A \,b^{3} e \,x^{3}+9 B a \,b^{2} e \,x^{3}-7 B \,b^{3} d \,x^{3}+3 A a \,b^{2} e \,x^{2}-9 A \,b^{3} d \,x^{2}+18 B \,a^{2} b e \,x^{2}-12 B a \,b^{2} d \,x^{2}+12 A \,a^{2} b e x -18 A a \,b^{2} d x +9 B \,a^{3} e x -3 B \,a^{2} b d x +7 a^{3} A e -9 A \,a^{2} b d +2 B \,a^{3} d \right ) \left (b x +a \right )^{\frac {3}{2}}}{63 \left (e x +d \right )^{\frac {9}{2}} \left (a e -d b \right )^{2}}\) | \(159\) |
Input:
int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(11/2),x,method=_RETURNVERBOSE)
Output:
-2/63*(b*x+a)^(7/2)*(-2*A*b*e*x+9*B*a*e*x-7*B*b*d*x+7*A*a*e-9*A*b*d+2*B*a* d)/(e*x+d)^(9/2)/(a^2*e^2-2*a*b*d*e+b^2*d^2)
Leaf count of result is larger than twice the leaf count of optimal. 391 vs. \(2 (83) = 166\).
Time = 28.89 (sec) , antiderivative size = 391, normalized size of antiderivative = 4.12 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{11/2}} \, dx=-\frac {2 \, {\left (7 \, A a^{4} e - {\left (7 \, B b^{4} d - {\left (9 \, B a b^{3} - 2 \, A b^{4}\right )} e\right )} x^{4} - {\left ({\left (19 \, B a b^{3} + 9 \, A b^{4}\right )} d - {\left (27 \, B a^{2} b^{2} + A a b^{3}\right )} e\right )} x^{3} - 3 \, {\left ({\left (5 \, B a^{2} b^{2} + 9 \, A a b^{3}\right )} d - {\left (9 \, B a^{3} b + 5 \, A a^{2} b^{2}\right )} e\right )} x^{2} + {\left (2 \, B a^{4} - 9 \, A a^{3} b\right )} d - {\left ({\left (B a^{3} b + 27 \, A a^{2} b^{2}\right )} d - {\left (9 \, B a^{4} + 19 \, A a^{3} b\right )} e\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{63 \, {\left (b^{2} d^{7} - 2 \, a b d^{6} e + a^{2} d^{5} e^{2} + {\left (b^{2} d^{2} e^{5} - 2 \, a b d e^{6} + a^{2} e^{7}\right )} x^{5} + 5 \, {\left (b^{2} d^{3} e^{4} - 2 \, a b d^{2} e^{5} + a^{2} d e^{6}\right )} x^{4} + 10 \, {\left (b^{2} d^{4} e^{3} - 2 \, a b d^{3} e^{4} + a^{2} d^{2} e^{5}\right )} x^{3} + 10 \, {\left (b^{2} d^{5} e^{2} - 2 \, a b d^{4} e^{3} + a^{2} d^{3} e^{4}\right )} x^{2} + 5 \, {\left (b^{2} d^{6} e - 2 \, a b d^{5} e^{2} + a^{2} d^{4} e^{3}\right )} x\right )}} \] Input:
integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(11/2),x, algorithm="fricas")
Output:
-2/63*(7*A*a^4*e - (7*B*b^4*d - (9*B*a*b^3 - 2*A*b^4)*e)*x^4 - ((19*B*a*b^ 3 + 9*A*b^4)*d - (27*B*a^2*b^2 + A*a*b^3)*e)*x^3 - 3*((5*B*a^2*b^2 + 9*A*a *b^3)*d - (9*B*a^3*b + 5*A*a^2*b^2)*e)*x^2 + (2*B*a^4 - 9*A*a^3*b)*d - ((B *a^3*b + 27*A*a^2*b^2)*d - (9*B*a^4 + 19*A*a^3*b)*e)*x)*sqrt(b*x + a)*sqrt (e*x + d)/(b^2*d^7 - 2*a*b*d^6*e + a^2*d^5*e^2 + (b^2*d^2*e^5 - 2*a*b*d*e^ 6 + a^2*e^7)*x^5 + 5*(b^2*d^3*e^4 - 2*a*b*d^2*e^5 + a^2*d*e^6)*x^4 + 10*(b ^2*d^4*e^3 - 2*a*b*d^3*e^4 + a^2*d^2*e^5)*x^3 + 10*(b^2*d^5*e^2 - 2*a*b*d^ 4*e^3 + a^2*d^3*e^4)*x^2 + 5*(b^2*d^6*e - 2*a*b*d^5*e^2 + a^2*d^4*e^3)*x)
\[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{11/2}} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {11}{2}}}\, dx \] Input:
integrate((b*x+a)**(5/2)*(B*x+A)/(e*x+d)**(11/2),x)
Output:
Integral((A + B*x)*(a + b*x)**(5/2)/(d + e*x)**(11/2), x)
Exception generated. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{11/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(11/2),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*(a*e-b*d)>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (83) = 166\).
Time = 0.56 (sec) , antiderivative size = 379, normalized size of antiderivative = 3.99 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{11/2}} \, dx=\frac {2 \, {\left (b x + a\right )}^{\frac {7}{2}} {\left (\frac {{\left (7 \, B b^{12} d^{3} e^{4} {\left | b \right |} - 23 \, B a b^{11} d^{2} e^{5} {\left | b \right |} + 2 \, A b^{12} d^{2} e^{5} {\left | b \right |} + 25 \, B a^{2} b^{10} d e^{6} {\left | b \right |} - 4 \, A a b^{11} d e^{6} {\left | b \right |} - 9 \, B a^{3} b^{9} e^{7} {\left | b \right |} + 2 \, A a^{2} b^{10} e^{7} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}} - \frac {9 \, {\left (B a b^{12} d^{3} e^{4} {\left | b \right |} - A b^{13} d^{3} e^{4} {\left | b \right |} - 3 \, B a^{2} b^{11} d^{2} e^{5} {\left | b \right |} + 3 \, A a b^{12} d^{2} e^{5} {\left | b \right |} + 3 \, B a^{3} b^{10} d e^{6} {\left | b \right |} - 3 \, A a^{2} b^{11} d e^{6} {\left | b \right |} - B a^{4} b^{9} e^{7} {\left | b \right |} + A a^{3} b^{10} e^{7} {\left | b \right |}\right )}}{b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}}\right )}}{63 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {9}{2}}} \] Input:
integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(11/2),x, algorithm="giac")
Output:
2/63*(b*x + a)^(7/2)*((7*B*b^12*d^3*e^4*abs(b) - 23*B*a*b^11*d^2*e^5*abs(b ) + 2*A*b^12*d^2*e^5*abs(b) + 25*B*a^2*b^10*d*e^6*abs(b) - 4*A*a*b^11*d*e^ 6*abs(b) - 9*B*a^3*b^9*e^7*abs(b) + 2*A*a^2*b^10*e^7*abs(b))*(b*x + a)/(b^ 6*d^4*e^4 - 4*a*b^5*d^3*e^5 + 6*a^2*b^4*d^2*e^6 - 4*a^3*b^3*d*e^7 + a^4*b^ 2*e^8) - 9*(B*a*b^12*d^3*e^4*abs(b) - A*b^13*d^3*e^4*abs(b) - 3*B*a^2*b^11 *d^2*e^5*abs(b) + 3*A*a*b^12*d^2*e^5*abs(b) + 3*B*a^3*b^10*d*e^6*abs(b) - 3*A*a^2*b^11*d*e^6*abs(b) - B*a^4*b^9*e^7*abs(b) + A*a^3*b^10*e^7*abs(b))/ (b^6*d^4*e^4 - 4*a*b^5*d^3*e^5 + 6*a^2*b^4*d^2*e^6 - 4*a^3*b^3*d*e^7 + a^4 *b^2*e^8))/(b^2*d + (b*x + a)*b*e - a*b*e)^(9/2)
Time = 1.70 (sec) , antiderivative size = 325, normalized size of antiderivative = 3.42 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{11/2}} \, dx=\frac {\sqrt {d+e\,x}\,\left (\frac {x^3\,\sqrt {a+b\,x}\,\left (18\,A\,b^4\,d-2\,A\,a\,b^3\,e+38\,B\,a\,b^3\,d-54\,B\,a^2\,b^2\,e\right )}{63\,e^5\,{\left (a\,e-b\,d\right )}^2}-\frac {\sqrt {a+b\,x}\,\left (14\,A\,a^4\,e+4\,B\,a^4\,d-18\,A\,a^3\,b\,d\right )}{63\,e^5\,{\left (a\,e-b\,d\right )}^2}+\frac {x^4\,\sqrt {a+b\,x}\,\left (4\,A\,b^4\,e+14\,B\,b^4\,d-18\,B\,a\,b^3\,e\right )}{63\,e^5\,{\left (a\,e-b\,d\right )}^2}-\frac {x\,\sqrt {a+b\,x}\,\left (18\,B\,a^4\,e+38\,A\,a^3\,b\,e-2\,B\,a^3\,b\,d-54\,A\,a^2\,b^2\,d\right )}{63\,e^5\,{\left (a\,e-b\,d\right )}^2}+\frac {2\,a\,b\,x^2\,\sqrt {a+b\,x}\,\left (9\,A\,b^2\,d-9\,B\,a^2\,e-5\,A\,a\,b\,e+5\,B\,a\,b\,d\right )}{21\,e^5\,{\left (a\,e-b\,d\right )}^2}\right )}{x^5+\frac {d^5}{e^5}+\frac {5\,d\,x^4}{e}+\frac {5\,d^4\,x}{e^4}+\frac {10\,d^2\,x^3}{e^2}+\frac {10\,d^3\,x^2}{e^3}} \] Input:
int(((A + B*x)*(a + b*x)^(5/2))/(d + e*x)^(11/2),x)
Output:
((d + e*x)^(1/2)*((x^3*(a + b*x)^(1/2)*(18*A*b^4*d - 2*A*a*b^3*e + 38*B*a* b^3*d - 54*B*a^2*b^2*e))/(63*e^5*(a*e - b*d)^2) - ((a + b*x)^(1/2)*(14*A*a ^4*e + 4*B*a^4*d - 18*A*a^3*b*d))/(63*e^5*(a*e - b*d)^2) + (x^4*(a + b*x)^ (1/2)*(4*A*b^4*e + 14*B*b^4*d - 18*B*a*b^3*e))/(63*e^5*(a*e - b*d)^2) - (x *(a + b*x)^(1/2)*(18*B*a^4*e + 38*A*a^3*b*e - 2*B*a^3*b*d - 54*A*a^2*b^2*d ))/(63*e^5*(a*e - b*d)^2) + (2*a*b*x^2*(a + b*x)^(1/2)*(9*A*b^2*d - 9*B*a^ 2*e - 5*A*a*b*e + 5*B*a*b*d))/(21*e^5*(a*e - b*d)^2)))/(x^5 + d^5/e^5 + (5 *d*x^4)/e + (5*d^4*x)/e^4 + (10*d^2*x^3)/e^2 + (10*d^3*x^2)/e^3)
Time = 2.29 (sec) , antiderivative size = 335, normalized size of antiderivative = 3.53 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{11/2}} \, dx=\frac {-\frac {2 \sqrt {e x +d}\, \sqrt {b x +a}\, a^{4} e^{5}}{9}-\frac {8 \sqrt {e x +d}\, \sqrt {b x +a}\, a^{3} b \,e^{5} x}{9}-\frac {4 \sqrt {e x +d}\, \sqrt {b x +a}\, a^{2} b^{2} e^{5} x^{2}}{3}-\frac {8 \sqrt {e x +d}\, \sqrt {b x +a}\, a \,b^{3} e^{5} x^{3}}{9}-\frac {2 \sqrt {e x +d}\, \sqrt {b x +a}\, b^{4} e^{5} x^{4}}{9}-\frac {2 \sqrt {e}\, \sqrt {b}\, b^{4} d^{5}}{9}-\frac {10 \sqrt {e}\, \sqrt {b}\, b^{4} d^{4} e x}{9}-\frac {20 \sqrt {e}\, \sqrt {b}\, b^{4} d^{3} e^{2} x^{2}}{9}-\frac {20 \sqrt {e}\, \sqrt {b}\, b^{4} d^{2} e^{3} x^{3}}{9}-\frac {10 \sqrt {e}\, \sqrt {b}\, b^{4} d \,e^{4} x^{4}}{9}-\frac {2 \sqrt {e}\, \sqrt {b}\, b^{4} e^{5} x^{5}}{9}}{e^{5} \left (a \,e^{6} x^{5}-b d \,e^{5} x^{5}+5 a d \,e^{5} x^{4}-5 b \,d^{2} e^{4} x^{4}+10 a \,d^{2} e^{4} x^{3}-10 b \,d^{3} e^{3} x^{3}+10 a \,d^{3} e^{3} x^{2}-10 b \,d^{4} e^{2} x^{2}+5 a \,d^{4} e^{2} x -5 b \,d^{5} e x +a \,d^{5} e -b \,d^{6}\right )} \] Input:
int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(11/2),x)
Output:
(2*( - sqrt(d + e*x)*sqrt(a + b*x)*a**4*e**5 - 4*sqrt(d + e*x)*sqrt(a + b* x)*a**3*b*e**5*x - 6*sqrt(d + e*x)*sqrt(a + b*x)*a**2*b**2*e**5*x**2 - 4*s qrt(d + e*x)*sqrt(a + b*x)*a*b**3*e**5*x**3 - sqrt(d + e*x)*sqrt(a + b*x)* b**4*e**5*x**4 - sqrt(e)*sqrt(b)*b**4*d**5 - 5*sqrt(e)*sqrt(b)*b**4*d**4*e *x - 10*sqrt(e)*sqrt(b)*b**4*d**3*e**2*x**2 - 10*sqrt(e)*sqrt(b)*b**4*d**2 *e**3*x**3 - 5*sqrt(e)*sqrt(b)*b**4*d*e**4*x**4 - sqrt(e)*sqrt(b)*b**4*e** 5*x**5))/(9*e**5*(a*d**5*e + 5*a*d**4*e**2*x + 10*a*d**3*e**3*x**2 + 10*a* d**2*e**4*x**3 + 5*a*d*e**5*x**4 + a*e**6*x**5 - b*d**6 - 5*b*d**5*e*x - 1 0*b*d**4*e**2*x**2 - 10*b*d**3*e**3*x**3 - 5*b*d**2*e**4*x**4 - b*d*e**5*x **5))