Integrand size = 24, antiderivative size = 145 \[ \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx=-\frac {2 (B d-A e) \sqrt {a+b x}}{5 e (b d-a e) (d+e x)^{5/2}}+\frac {2 (b B d+4 A b e-5 a B e) \sqrt {a+b x}}{15 e (b d-a e)^2 (d+e x)^{3/2}}+\frac {4 b (b B d+4 A b e-5 a B e) \sqrt {a+b x}}{15 e (b d-a e)^3 \sqrt {d+e x}} \]
-2/5*(-A*e+B*d)*(b*x+a)^(1/2)/e/(-a*e+b*d)/(e*x+d)^(5/2)+2/15*(4*A*b*e-5*B *a*e+B*b*d)*(b*x+a)^(1/2)/e/(-a*e+b*d)^2/(e*x+d)^(3/2)+4/15*b*(4*A*b*e-5*B *a*e+B*b*d)*(b*x+a)^(1/2)/e/(-a*e+b*d)^3/(e*x+d)^(1/2)
Time = 0.16 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx=\frac {2 \sqrt {a+b x} \left (15 A b^2-15 a b B-\frac {3 B d e (a+b x)^2}{(d+e x)^2}+\frac {3 A e^2 (a+b x)^2}{(d+e x)^2}+\frac {5 b B d (a+b x)}{d+e x}-\frac {10 A b e (a+b x)}{d+e x}+\frac {5 a B e (a+b x)}{d+e x}\right )}{15 (b d-a e)^3 \sqrt {d+e x}} \]
(2*Sqrt[a + b*x]*(15*A*b^2 - 15*a*b*B - (3*B*d*e*(a + b*x)^2)/(d + e*x)^2 + (3*A*e^2*(a + b*x)^2)/(d + e*x)^2 + (5*b*B*d*(a + b*x))/(d + e*x) - (10* A*b*e*(a + b*x))/(d + e*x) + (5*a*B*e*(a + b*x))/(d + e*x)))/(15*(b*d - a* e)^3*Sqrt[d + e*x])
Time = 0.23 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {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.
\(\displaystyle \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {(-5 a B e+4 A b e+b B d) \int \frac {1}{\sqrt {a+b x} (d+e x)^{5/2}}dx}{5 e (b d-a e)}-\frac {2 \sqrt {a+b x} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {(-5 a B e+4 A b e+b B d) \left (\frac {2 b \int \frac {1}{\sqrt {a+b x} (d+e x)^{3/2}}dx}{3 (b d-a e)}+\frac {2 \sqrt {a+b x}}{3 (d+e x)^{3/2} (b d-a e)}\right )}{5 e (b d-a e)}-\frac {2 \sqrt {a+b x} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {\left (\frac {4 b \sqrt {a+b x}}{3 \sqrt {d+e x} (b d-a e)^2}+\frac {2 \sqrt {a+b x}}{3 (d+e x)^{3/2} (b d-a e)}\right ) (-5 a B e+4 A b e+b B d)}{5 e (b d-a e)}-\frac {2 \sqrt {a+b x} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)}\) |
(-2*(B*d - A*e)*Sqrt[a + b*x])/(5*e*(b*d - a*e)*(d + e*x)^(5/2)) + ((b*B*d + 4*A*b*e - 5*a*B*e)*((2*Sqrt[a + b*x])/(3*(b*d - a*e)*(d + e*x)^(3/2)) + (4*b*Sqrt[a + b*x])/(3*(b*d - a*e)^2*Sqrt[d + e*x])))/(5*e*(b*d - a*e))
3.23.42.3.1 Defintions of rubi rules used
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]))))
Time = 1.09 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.03
method | result | size |
default | \(-\frac {2 \sqrt {b x +a}\, \left (8 A \,b^{2} e^{2} x^{2}-10 B a b \,e^{2} x^{2}+2 B \,b^{2} d e \,x^{2}-4 A a b \,e^{2} x +20 A \,b^{2} d e x +5 B \,a^{2} e^{2} x -26 B a b d e x +5 b^{2} B \,d^{2} x +3 a^{2} A \,e^{2}-10 A a b d e +15 A \,b^{2} d^{2}+2 B \,a^{2} d e -10 B a b \,d^{2}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} \left (a e -b d \right )^{3}}\) | \(149\) |
gosper | \(-\frac {2 \sqrt {b x +a}\, \left (8 A \,b^{2} e^{2} x^{2}-10 B a b \,e^{2} x^{2}+2 B \,b^{2} d e \,x^{2}-4 A a b \,e^{2} x +20 A \,b^{2} d e x +5 B \,a^{2} e^{2} x -26 B a b d e x +5 b^{2} B \,d^{2} x +3 a^{2} A \,e^{2}-10 A a b d e +15 A \,b^{2} d^{2}+2 B \,a^{2} d e -10 B a b \,d^{2}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}\) | \(177\) |
-2/15*(b*x+a)^(1/2)*(8*A*b^2*e^2*x^2-10*B*a*b*e^2*x^2+2*B*b^2*d*e*x^2-4*A* a*b*e^2*x+20*A*b^2*d*e*x+5*B*a^2*e^2*x-26*B*a*b*d*e*x+5*B*b^2*d^2*x+3*A*a^ 2*e^2-10*A*a*b*d*e+15*A*b^2*d^2+2*B*a^2*d*e-10*B*a*b*d^2)/(e*x+d)^(5/2)/(a *e-b*d)^3
Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (127) = 254\).
Time = 2.04 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.18 \[ \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx=\frac {2 \, {\left (3 \, A a^{2} e^{2} - 5 \, {\left (2 \, B a b - 3 \, A b^{2}\right )} d^{2} + 2 \, {\left (B a^{2} - 5 \, A a b\right )} d e + 2 \, {\left (B b^{2} d e - {\left (5 \, B a b - 4 \, A b^{2}\right )} e^{2}\right )} x^{2} + {\left (5 \, B b^{2} d^{2} - 2 \, {\left (13 \, B a b - 10 \, A b^{2}\right )} d e + {\left (5 \, B a^{2} - 4 \, A a b\right )} e^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{15 \, {\left (b^{3} d^{6} - 3 \, a b^{2} d^{5} e + 3 \, a^{2} b d^{4} e^{2} - a^{3} d^{3} e^{3} + {\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} x^{3} + 3 \, {\left (b^{3} d^{4} e^{2} - 3 \, a b^{2} d^{3} e^{3} + 3 \, a^{2} b d^{2} e^{4} - a^{3} d e^{5}\right )} x^{2} + 3 \, {\left (b^{3} d^{5} e - 3 \, a b^{2} d^{4} e^{2} + 3 \, a^{2} b d^{3} e^{3} - a^{3} d^{2} e^{4}\right )} x\right )}} \]
2/15*(3*A*a^2*e^2 - 5*(2*B*a*b - 3*A*b^2)*d^2 + 2*(B*a^2 - 5*A*a*b)*d*e + 2*(B*b^2*d*e - (5*B*a*b - 4*A*b^2)*e^2)*x^2 + (5*B*b^2*d^2 - 2*(13*B*a*b - 10*A*b^2)*d*e + (5*B*a^2 - 4*A*a*b)*e^2)*x)*sqrt(b*x + a)*sqrt(e*x + d)/( b^3*d^6 - 3*a*b^2*d^5*e + 3*a^2*b*d^4*e^2 - a^3*d^3*e^3 + (b^3*d^3*e^3 - 3 *a*b^2*d^2*e^4 + 3*a^2*b*d*e^5 - a^3*e^6)*x^3 + 3*(b^3*d^4*e^2 - 3*a*b^2*d ^3*e^3 + 3*a^2*b*d^2*e^4 - a^3*d*e^5)*x^2 + 3*(b^3*d^5*e - 3*a*b^2*d^4*e^2 + 3*a^2*b*d^3*e^3 - a^3*d^2*e^4)*x)
\[ \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx=\int \frac {A + B x}{\sqrt {a + b x} \left (d + e x\right )^{\frac {7}{2}}}\, dx \]
Exception generated. \[ \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx=\text {Exception raised: ValueError} \]
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. 377 vs. \(2 (127) = 254\).
Time = 0.37 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.60 \[ \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx=\frac {2 \, {\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (B b^{6} d e^{3} {\left | b \right |} - 5 \, B a b^{5} e^{4} {\left | b \right |} + 4 \, A b^{6} e^{4} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{5} d^{3} e^{2} - 3 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}} + \frac {5 \, {\left (B b^{7} d^{2} e^{2} {\left | b \right |} - 6 \, B a b^{6} d e^{3} {\left | b \right |} + 4 \, A b^{7} d e^{3} {\left | b \right |} + 5 \, B a^{2} b^{5} e^{4} {\left | b \right |} - 4 \, A a b^{6} e^{4} {\left | b \right |}\right )}}{b^{5} d^{3} e^{2} - 3 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}}\right )} - \frac {15 \, {\left (B a b^{7} d^{2} e^{2} {\left | b \right |} - A b^{8} d^{2} e^{2} {\left | b \right |} - 2 \, B a^{2} b^{6} d e^{3} {\left | b \right |} + 2 \, A a b^{7} d e^{3} {\left | b \right |} + B a^{3} b^{5} e^{4} {\left | b \right |} - A a^{2} b^{6} e^{4} {\left | b \right |}\right )}}{b^{5} d^{3} e^{2} - 3 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}}\right )} \sqrt {b x + a}}{15 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {5}{2}}} \]
2/15*((b*x + a)*(2*(B*b^6*d*e^3*abs(b) - 5*B*a*b^5*e^4*abs(b) + 4*A*b^6*e^ 4*abs(b))*(b*x + a)/(b^5*d^3*e^2 - 3*a*b^4*d^2*e^3 + 3*a^2*b^3*d*e^4 - a^3 *b^2*e^5) + 5*(B*b^7*d^2*e^2*abs(b) - 6*B*a*b^6*d*e^3*abs(b) + 4*A*b^7*d*e ^3*abs(b) + 5*B*a^2*b^5*e^4*abs(b) - 4*A*a*b^6*e^4*abs(b))/(b^5*d^3*e^2 - 3*a*b^4*d^2*e^3 + 3*a^2*b^3*d*e^4 - a^3*b^2*e^5)) - 15*(B*a*b^7*d^2*e^2*ab s(b) - A*b^8*d^2*e^2*abs(b) - 2*B*a^2*b^6*d*e^3*abs(b) + 2*A*a*b^7*d*e^3*a bs(b) + B*a^3*b^5*e^4*abs(b) - A*a^2*b^6*e^4*abs(b))/(b^5*d^3*e^2 - 3*a*b^ 4*d^2*e^3 + 3*a^2*b^3*d*e^4 - a^3*b^2*e^5))*sqrt(b*x + a)/(b^2*d + (b*x + a)*b*e - a*b*e)^(5/2)
Time = 2.65 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.95 \[ \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx=-\frac {\sqrt {d+e\,x}\,\left (\frac {4\,B\,a^3\,d\,e+6\,A\,a^3\,e^2-20\,B\,a^2\,b\,d^2-20\,A\,a^2\,b\,d\,e+30\,A\,a\,b^2\,d^2}{15\,e^3\,{\left (a\,e-b\,d\right )}^3}+\frac {x\,\left (10\,B\,a^3\,e^2-48\,B\,a^2\,b\,d\,e-2\,A\,a^2\,b\,e^2-10\,B\,a\,b^2\,d^2+20\,A\,a\,b^2\,d\,e+30\,A\,b^3\,d^2\right )}{15\,e^3\,{\left (a\,e-b\,d\right )}^3}+\frac {4\,b^2\,x^3\,\left (4\,A\,b\,e-5\,B\,a\,e+B\,b\,d\right )}{15\,e^2\,{\left (a\,e-b\,d\right )}^3}+\frac {2\,b\,x^2\,\left (a\,e+5\,b\,d\right )\,\left (4\,A\,b\,e-5\,B\,a\,e+B\,b\,d\right )}{15\,e^3\,{\left (a\,e-b\,d\right )}^3}\right )}{x^3\,\sqrt {a+b\,x}+\frac {d^3\,\sqrt {a+b\,x}}{e^3}+\frac {3\,d\,x^2\,\sqrt {a+b\,x}}{e}+\frac {3\,d^2\,x\,\sqrt {a+b\,x}}{e^2}} \]
-((d + e*x)^(1/2)*((6*A*a^3*e^2 + 4*B*a^3*d*e + 30*A*a*b^2*d^2 - 20*B*a^2* b*d^2 - 20*A*a^2*b*d*e)/(15*e^3*(a*e - b*d)^3) + (x*(30*A*b^3*d^2 + 10*B*a ^3*e^2 - 2*A*a^2*b*e^2 - 10*B*a*b^2*d^2 + 20*A*a*b^2*d*e - 48*B*a^2*b*d*e) )/(15*e^3*(a*e - b*d)^3) + (4*b^2*x^3*(4*A*b*e - 5*B*a*e + B*b*d))/(15*e^2 *(a*e - b*d)^3) + (2*b*x^2*(a*e + 5*b*d)*(4*A*b*e - 5*B*a*e + B*b*d))/(15* e^3*(a*e - b*d)^3)))/(x^3*(a + b*x)^(1/2) + (d^3*(a + b*x)^(1/2))/e^3 + (3 *d*x^2*(a + b*x)^(1/2))/e + (3*d^2*x*(a + b*x)^(1/2))/e^2)