Integrand size = 24, antiderivative size = 193 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{\sqrt {d+e x}} \, dx=\frac {(b d-a e) (5 b B d-6 A b e+a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 b e^3}-\frac {(5 b B d-6 A b e+a B e) (a+b x)^{3/2} \sqrt {d+e x}}{12 b e^2}+\frac {B (a+b x)^{5/2} \sqrt {d+e x}}{3 b e}-\frac {(b d-a e)^2 (5 b B d-6 A b e+a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{3/2} e^{7/2}} \]
-1/8*(-a*e+b*d)^2*(-6*A*b*e+B*a*e+5*B*b*d)*arctanh(e^(1/2)*(b*x+a)^(1/2)/b ^(1/2)/(e*x+d)^(1/2))/b^(3/2)/e^(7/2)-1/12*(-6*A*b*e+B*a*e+5*B*b*d)*(b*x+a )^(3/2)*(e*x+d)^(1/2)/b/e^2+1/3*B*(b*x+a)^(5/2)*(e*x+d)^(1/2)/b/e+1/8*(-a* e+b*d)*(-6*A*b*e+B*a*e+5*B*b*d)*(b*x+a)^(1/2)*(e*x+d)^(1/2)/b/e^3
Time = 0.42 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{\sqrt {d+e x}} \, dx=\frac {\sqrt {a+b x} \sqrt {d+e x} \left (3 a^2 B e^2+2 a b e (-11 B d+15 A e+7 B e x)+b^2 \left (6 A e (-3 d+2 e x)+B \left (15 d^2-10 d e x+8 e^2 x^2\right )\right )\right )}{24 b e^3}-\frac {(b d-a e)^2 (5 b B d-6 A b e+a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{8 b^{3/2} e^{7/2}} \]
(Sqrt[a + b*x]*Sqrt[d + e*x]*(3*a^2*B*e^2 + 2*a*b*e*(-11*B*d + 15*A*e + 7* B*e*x) + b^2*(6*A*e*(-3*d + 2*e*x) + B*(15*d^2 - 10*d*e*x + 8*e^2*x^2))))/ (24*b*e^3) - ((b*d - a*e)^2*(5*b*B*d - 6*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*S qrt[d + e*x])/(Sqrt[e]*Sqrt[a + b*x])])/(8*b^(3/2)*e^(7/2))
Time = 0.25 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.88, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {90, 60, 60, 66, 221}
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.
| \(\displaystyle \int \frac {(a+b x)^{3/2} (A+B x)}{\sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {B (a+b x)^{5/2} \sqrt {d+e x}}{3 b e}-\frac {(a B e-6 A b e+5 b B d) \int \frac {(a+b x)^{3/2}}{\sqrt {d+e x}}dx}{6 b e}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {B (a+b x)^{5/2} \sqrt {d+e x}}{3 b e}-\frac {(a B e-6 A b e+5 b B d) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}}dx}{4 e}\right )}{6 b e}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {B (a+b x)^{5/2} \sqrt {d+e x}}{3 b e}-\frac {(a B e-6 A b e+5 b B d) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}}dx}{2 e}\right )}{4 e}\right )}{6 b e}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {B (a+b x)^{5/2} \sqrt {d+e x}}{3 b e}-\frac {(a B e-6 A b e+5 b B d) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \int \frac {1}{b-\frac {e (a+b x)}{d+e x}}d\frac {\sqrt {a+b x}}{\sqrt {d+e x}}}{e}\right )}{4 e}\right )}{6 b e}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {B (a+b x)^{5/2} \sqrt {d+e x}}{3 b e}-\frac {(a B e-6 A b e+5 b B d) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} e^{3/2}}\right )}{4 e}\right )}{6 b e}\) |
(B*(a + b*x)^(5/2)*Sqrt[d + e*x])/(3*b*e) - ((5*b*B*d - 6*A*b*e + a*B*e)*( ((a + b*x)^(3/2)*Sqrt[d + e*x])/(2*e) - (3*(b*d - a*e)*((Sqrt[a + b*x]*Sqr t[d + e*x])/e - ((b*d - a*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt [d + e*x])])/(Sqrt[b]*e^(3/2))))/(4*e)))/(6*b*e)
3.23.14.3.1 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/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Leaf count of result is larger than twice the leaf count of optimal. \(635\) vs. \(2(161)=322\).
Time = 1.06 (sec) , antiderivative size = 636, normalized size of antiderivative = 3.30
| method | result | size |
| default | \(\frac {\sqrt {b x +a}\, \sqrt {e x +d}\, \left (16 B \,b^{2} e^{2} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+18 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} b \,e^{3}-36 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{2} d \,e^{2}+18 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{3} d^{2} e +24 A \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{2} e^{2} x -3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{3} e^{3}-9 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} b d \,e^{2}+27 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{2} d^{2} e -15 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{3} d^{3}+28 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a b \,e^{2} x -20 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{2} d e x +60 A \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a b \,e^{2}-36 A \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{2} d e +6 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a^{2} e^{2}-44 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a b d e +30 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{2} d^{2}\right )}{48 b \,e^{3} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}\) | \(636\) |
1/48*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(16*B*b^2*e^2*x^2*((b*x+a)*(e*x+d))^(1/2) *(b*e)^(1/2)+18*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a* e+b*d)/(b*e)^(1/2))*a^2*b*e^3-36*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/ 2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^2*d*e^2+18*A*ln(1/2*(2*b*e*x+2*(( b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^3*d^2*e+24*A*((b *x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*b^2*e^2*x-3*B*ln(1/2*(2*b*e*x+2*((b*x+a)* (e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*e^3-9*B*ln(1/2*(2*b*e *x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b*d*e^2 +27*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e) ^(1/2))*a*b^2*d^2*e-15*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^( 1/2)+a*e+b*d)/(b*e)^(1/2))*b^3*d^3+28*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2 )*a*b*e^2*x-20*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*b^2*d*e*x+60*A*((b*x+ a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a*b*e^2-36*A*((b*x+a)*(e*x+d))^(1/2)*(b*e)^( 1/2)*b^2*d*e+6*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a^2*e^2-44*B*((b*x+a) *(e*x+d))^(1/2)*(b*e)^(1/2)*a*b*d*e+30*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/ 2)*b^2*d^2)/b/e^3/((b*x+a)*(e*x+d))^(1/2)/(b*e)^(1/2)
Time = 0.28 (sec) , antiderivative size = 540, normalized size of antiderivative = 2.80 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{\sqrt {d+e x}} \, dx=\left [-\frac {3 \, {\left (5 \, B b^{3} d^{3} - 3 \, {\left (3 \, B a b^{2} + 2 \, A b^{3}\right )} d^{2} e + 3 \, {\left (B a^{2} b + 4 \, A a b^{2}\right )} d e^{2} + {\left (B a^{3} - 6 \, A a^{2} b\right )} e^{3}\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) - 4 \, {\left (8 \, B b^{3} e^{3} x^{2} + 15 \, B b^{3} d^{2} e - 2 \, {\left (11 \, B a b^{2} + 9 \, A b^{3}\right )} d e^{2} + 3 \, {\left (B a^{2} b + 10 \, A a b^{2}\right )} e^{3} - 2 \, {\left (5 \, B b^{3} d e^{2} - {\left (7 \, B a b^{2} + 6 \, A b^{3}\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{96 \, b^{2} e^{4}}, \frac {3 \, {\left (5 \, B b^{3} d^{3} - 3 \, {\left (3 \, B a b^{2} + 2 \, A b^{3}\right )} d^{2} e + 3 \, {\left (B a^{2} b + 4 \, A a b^{2}\right )} d e^{2} + {\left (B a^{3} - 6 \, A a^{2} b\right )} e^{3}\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) + 2 \, {\left (8 \, B b^{3} e^{3} x^{2} + 15 \, B b^{3} d^{2} e - 2 \, {\left (11 \, B a b^{2} + 9 \, A b^{3}\right )} d e^{2} + 3 \, {\left (B a^{2} b + 10 \, A a b^{2}\right )} e^{3} - 2 \, {\left (5 \, B b^{3} d e^{2} - {\left (7 \, B a b^{2} + 6 \, A b^{3}\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{48 \, b^{2} e^{4}}\right ] \]
[-1/96*(3*(5*B*b^3*d^3 - 3*(3*B*a*b^2 + 2*A*b^3)*d^2*e + 3*(B*a^2*b + 4*A* a*b^2)*d*e^2 + (B*a^3 - 6*A*a^2*b)*e^3)*sqrt(b*e)*log(8*b^2*e^2*x^2 + b^2* d^2 + 6*a*b*d*e + a^2*e^2 + 4*(2*b*e*x + b*d + a*e)*sqrt(b*e)*sqrt(b*x + a )*sqrt(e*x + d) + 8*(b^2*d*e + a*b*e^2)*x) - 4*(8*B*b^3*e^3*x^2 + 15*B*b^3 *d^2*e - 2*(11*B*a*b^2 + 9*A*b^3)*d*e^2 + 3*(B*a^2*b + 10*A*a*b^2)*e^3 - 2 *(5*B*b^3*d*e^2 - (7*B*a*b^2 + 6*A*b^3)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d ))/(b^2*e^4), 1/48*(3*(5*B*b^3*d^3 - 3*(3*B*a*b^2 + 2*A*b^3)*d^2*e + 3*(B* a^2*b + 4*A*a*b^2)*d*e^2 + (B*a^3 - 6*A*a^2*b)*e^3)*sqrt(-b*e)*arctan(1/2* (2*b*e*x + b*d + a*e)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(e*x + d)/(b^2*e^2*x^2 + a*b*d*e + (b^2*d*e + a*b*e^2)*x)) + 2*(8*B*b^3*e^3*x^2 + 15*B*b^3*d^2*e - 2*(11*B*a*b^2 + 9*A*b^3)*d*e^2 + 3*(B*a^2*b + 10*A*a*b^2)*e^3 - 2*(5*B*b ^3*d*e^2 - (7*B*a*b^2 + 6*A*b^3)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^2 *e^4)]
\[ \int \frac {(a+b x)^{3/2} (A+B x)}{\sqrt {d+e x}} \, dx=\int \frac {\left (A + B x\right ) \left (a + b x\right )^{\frac {3}{2}}}{\sqrt {d + e x}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{3/2} (A+B x)}{\sqrt {d+e x}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for m ore detail
Time = 0.33 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.47 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{\sqrt {d+e x}} \, dx=\frac {{\left (\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )} B}{b^{2} e} - \frac {5 \, B b^{3} d e^{3} + B a b^{2} e^{4} - 6 \, A b^{3} e^{4}}{b^{4} e^{5}}\right )} + \frac {3 \, {\left (5 \, B b^{4} d^{2} e^{2} - 4 \, B a b^{3} d e^{3} - 6 \, A b^{4} d e^{3} - B a^{2} b^{2} e^{4} + 6 \, A a b^{3} e^{4}\right )}}{b^{4} e^{5}}\right )} + \frac {3 \, {\left (5 \, B b^{3} d^{3} - 9 \, B a b^{2} d^{2} e - 6 \, A b^{3} d^{2} e + 3 \, B a^{2} b d e^{2} + 12 \, A a b^{2} d e^{2} + B a^{3} e^{3} - 6 \, A a^{2} b e^{3}\right )} \log \left ({\left | -\sqrt {b e} \sqrt {b x + a} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b e} b e^{3}}\right )} b}{24 \, {\left | b \right |}} \]
1/24*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b *x + a)*B/(b^2*e) - (5*B*b^3*d*e^3 + B*a*b^2*e^4 - 6*A*b^3*e^4)/(b^4*e^5)) + 3*(5*B*b^4*d^2*e^2 - 4*B*a*b^3*d*e^3 - 6*A*b^4*d*e^3 - B*a^2*b^2*e^4 + 6*A*a*b^3*e^4)/(b^4*e^5)) + 3*(5*B*b^3*d^3 - 9*B*a*b^2*d^2*e - 6*A*b^3*d^2 *e + 3*B*a^2*b*d*e^2 + 12*A*a*b^2*d*e^2 + B*a^3*e^3 - 6*A*a^2*b*e^3)*log(a bs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/(sqrt( b*e)*b*e^3))*b/abs(b)
Timed out. \[ \int \frac {(a+b x)^{3/2} (A+B x)}{\sqrt {d+e x}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{3/2}}{\sqrt {d+e\,x}} \,d x \]