Integrand size = 19, antiderivative size = 186 \[ \int \sqrt {a+b x} (c+d x)^{5/2} \, dx=\frac {5 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 b^3 d}+\frac {5 (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{32 b^3}+\frac {5 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2}+\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}-\frac {5 (b c-a d)^4 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{7/2} d^{3/2}} \]
5/24*(-a*d+b*c)*(b*x+a)^(3/2)*(d*x+c)^(3/2)/b^2+1/4*(b*x+a)^(3/2)*(d*x+c)^ (5/2)/b-5/64*(-a*d+b*c)^4*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1 /2))/b^(7/2)/d^(3/2)+5/32*(-a*d+b*c)^2*(b*x+a)^(3/2)*(d*x+c)^(1/2)/b^3+5/6 4*(-a*d+b*c)^3*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^3/d
Time = 0.27 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.90 \[ \int \sqrt {a+b x} (c+d x)^{5/2} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (15 a^3 d^3-5 a^2 b d^2 (11 c+2 d x)+a b^2 d \left (73 c^2+36 c d x+8 d^2 x^2\right )+b^3 \left (15 c^3+118 c^2 d x+136 c d^2 x^2+48 d^3 x^3\right )\right )}{192 b^3 d}-\frac {5 (b c-a d)^4 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{64 b^{7/2} d^{3/2}} \]
(Sqrt[a + b*x]*Sqrt[c + d*x]*(15*a^3*d^3 - 5*a^2*b*d^2*(11*c + 2*d*x) + a* b^2*d*(73*c^2 + 36*c*d*x + 8*d^2*x^2) + b^3*(15*c^3 + 118*c^2*d*x + 136*c* d^2*x^2 + 48*d^3*x^3)))/(192*b^3*d) - (5*(b*c - a*d)^4*ArcTanh[(Sqrt[b]*Sq rt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(64*b^(7/2)*d^(3/2))
Time = 0.25 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {60, 60, 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.
| \(\displaystyle \int \sqrt {a+b x} (c+d x)^{5/2} \, dx\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {5 (b c-a d) \int \sqrt {a+b x} (c+d x)^{3/2}dx}{8 b}+\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {5 (b c-a d) \left (\frac {(b c-a d) \int \sqrt {a+b x} \sqrt {c+d x}dx}{2 b}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b}\right )}{8 b}+\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {5 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}}dx}{4 b}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b}\right )}{2 b}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b}\right )}{8 b}+\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {5 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 d}\right )}{4 b}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b}\right )}{2 b}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b}\right )}{8 b}+\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {5 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{d}\right )}{4 b}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b}\right )}{2 b}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b}\right )}{8 b}+\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {5 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{3/2}}\right )}{4 b}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b}\right )}{2 b}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b}\right )}{8 b}+\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}\) |
((a + b*x)^(3/2)*(c + d*x)^(5/2))/(4*b) + (5*(b*c - a*d)*(((a + b*x)^(3/2) *(c + d*x)^(3/2))/(3*b) + ((b*c - a*d)*(((a + b*x)^(3/2)*Sqrt[c + d*x])/(2 *b) + ((b*c - a*d)*((Sqrt[a + b*x]*Sqrt[c + d*x])/d - ((b*c - a*d)*ArcTanh [(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqrt[b]*d^(3/2))))/(4* b)))/(2*b)))/(8*b)
3.15.83.3.1 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_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Time = 0.26 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.11
| method | result | size |
| default | \(\frac {\sqrt {b x +a}\, \left (d x +c \right )^{\frac {7}{2}}}{4 d}-\frac {\left (-a d +b c \right ) \left (\frac {\left (d x +c \right )^{\frac {5}{2}} \sqrt {b x +a}}{3 b}-\frac {5 \left (a d -b c \right ) \left (\frac {\left (d x +c \right )^{\frac {3}{2}} \sqrt {b x +a}}{2 b}-\frac {3 \left (a d -b c \right ) \left (\frac {\sqrt {b x +a}\, \sqrt {d x +c}}{b}-\frac {\left (a d -b c \right ) \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d x}{\sqrt {b d}}+\sqrt {b d \,x^{2}+\left (a d +b c \right ) x +a c}\right )}{2 b \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}}\right )}{4 b}\right )}{6 b}\right )}{8 d}\) | \(206\) |
1/4/d*(b*x+a)^(1/2)*(d*x+c)^(7/2)-1/8*(-a*d+b*c)/d*(1/3*(d*x+c)^(5/2)*(b*x +a)^(1/2)/b-5/6*(a*d-b*c)/b*(1/2*(d*x+c)^(3/2)*(b*x+a)^(1/2)/b-3/4*(a*d-b* c)/b*((b*x+a)^(1/2)*(d*x+c)^(1/2)/b-1/2*(a*d-b*c)/b*((b*x+a)*(d*x+c))^(1/2 )/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(b*d* x^2+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2))))
Time = 0.26 (sec) , antiderivative size = 540, normalized size of antiderivative = 2.90 \[ \int \sqrt {a+b x} (c+d x)^{5/2} \, dx=\left [\frac {15 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (48 \, b^{4} d^{4} x^{3} + 15 \, b^{4} c^{3} d + 73 \, a b^{3} c^{2} d^{2} - 55 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4} + 8 \, {\left (17 \, b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (59 \, b^{4} c^{2} d^{2} + 18 \, a b^{3} c d^{3} - 5 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, b^{4} d^{2}}, \frac {15 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (48 \, b^{4} d^{4} x^{3} + 15 \, b^{4} c^{3} d + 73 \, a b^{3} c^{2} d^{2} - 55 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4} + 8 \, {\left (17 \, b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (59 \, b^{4} c^{2} d^{2} + 18 \, a b^{3} c d^{3} - 5 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, b^{4} d^{2}}\right ] \]
[1/768*(15*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 - 4*( 2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 4*(48*b^4*d^4*x^3 + 15*b^4*c^3*d + 73*a*b^3*c^2*d^2 - 55*a^2 *b^2*c*d^3 + 15*a^3*b*d^4 + 8*(17*b^4*c*d^3 + a*b^3*d^4)*x^2 + 2*(59*b^4*c ^2*d^2 + 18*a*b^3*c*d^3 - 5*a^2*b^2*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/( b^4*d^2), 1/384*(15*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b *c*d^3 + a^4*d^4)*sqrt(-b*d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*s qrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x) ) + 2*(48*b^4*d^4*x^3 + 15*b^4*c^3*d + 73*a*b^3*c^2*d^2 - 55*a^2*b^2*c*d^3 + 15*a^3*b*d^4 + 8*(17*b^4*c*d^3 + a*b^3*d^4)*x^2 + 2*(59*b^4*c^2*d^2 + 1 8*a*b^3*c*d^3 - 5*a^2*b^2*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^4*d^2)]
\[ \int \sqrt {a+b x} (c+d x)^{5/2} \, dx=\int \sqrt {a + b x} \left (c + d x\right )^{\frac {5}{2}}\, dx \]
Exception generated. \[ \int \sqrt {a+b x} (c+d x)^{5/2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 1083 vs. \(2 (148) = 296\).
Time = 0.48 (sec) , antiderivative size = 1083, normalized size of antiderivative = 5.82 \[ \int \sqrt {a+b x} (c+d x)^{5/2} \, dx=\text {Too large to display} \]
-1/192*(192*((b^2*c - a*b*d)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/sqrt(b*d) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d )*sqrt(b*x + a))*a*c^2*abs(b)/b^2 - 16*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d )*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*c*d^3 - 13*a*b^5*d^4) /(b^7*d^4)) - 3*(b^7*c^2*d^2 + 2*a*b^6*c*d^3 - 11*a^2*b^5*d^4)/(b^7*d^4)) - 3*(b^3*c^3 + a*b^2*c^2*d + 3*a^2*b*c*d^2 - 5*a^3*d^3)*log(abs(-sqrt(b*d) *sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b*d^2))* c*d*abs(b)/b - 8*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b* x + a)*(4*(b*x + a)/b^2 + (b^6*c*d^3 - 13*a*b^5*d^4)/(b^7*d^4)) - 3*(b^7*c ^2*d^2 + 2*a*b^6*c*d^3 - 11*a^2*b^5*d^4)/(b^7*d^4)) - 3*(b^3*c^3 + a*b^2*c ^2*d + 3*a^2*b*c*d^2 - 5*a^3*d^3)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt( b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b*d^2))*a*d^2*abs(b)/b^2 - (sq rt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b ^3 + (b^12*c*d^5 - 25*a*b^11*d^6)/(b^14*d^6)) - (5*b^13*c^2*d^4 + 14*a*b^1 2*c*d^5 - 163*a^2*b^11*d^6)/(b^14*d^6)) + 3*(5*b^14*c^3*d^3 + 9*a*b^13*c^2 *d^4 + 15*a^2*b^12*c*d^5 - 93*a^3*b^11*d^6)/(b^14*d^6))*sqrt(b*x + a) + 3* (5*b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 35*a^4*d ^4)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d) ))/(sqrt(b*d)*b^2*d^3))*d^2*abs(b)/b - 48*(sqrt(b^2*c + (b*x + a)*b*d - a* b*d)*(2*b*x + 2*a + (b*c*d - 5*a*d^2)/d^2)*sqrt(b*x + a) + (b^3*c^2 + 2...
Timed out. \[ \int \sqrt {a+b x} (c+d x)^{5/2} \, dx=\int \sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/2} \,d x \]
Time = 0.02 (sec) , antiderivative size = 471, normalized size of antiderivative = 2.53 \[ \int \sqrt {a+b x} (c+d x)^{5/2} \, dx=\frac {15 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{3} b \,d^{4}-55 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} b^{2} c \,d^{3}-10 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} b^{2} d^{4} x +73 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{3} c^{2} d^{2}+36 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{3} c \,d^{3} x +8 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{3} d^{4} x^{2}+15 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{4} c^{3} d +118 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{4} c^{2} d^{2} x +136 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{4} c \,d^{3} x^{2}+48 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{4} d^{4} x^{3}-15 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{4} d^{4}+60 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{3} b c \,d^{3}-90 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} b^{2} c^{2} d^{2}+60 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a \,b^{3} c^{3} d -15 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{4} c^{4}}{192 b^{4} d^{2}} \]
(15*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b*d**4 - 55*sqrt(c + d*x)*sqrt(a + b* x)*a**2*b**2*c*d**3 - 10*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**2*d**4*x + 73 *sqrt(c + d*x)*sqrt(a + b*x)*a*b**3*c**2*d**2 + 36*sqrt(c + d*x)*sqrt(a + b*x)*a*b**3*c*d**3*x + 8*sqrt(c + d*x)*sqrt(a + b*x)*a*b**3*d**4*x**2 + 15 *sqrt(c + d*x)*sqrt(a + b*x)*b**4*c**3*d + 118*sqrt(c + d*x)*sqrt(a + b*x) *b**4*c**2*d**2*x + 136*sqrt(c + d*x)*sqrt(a + b*x)*b**4*c*d**3*x**2 + 48* sqrt(c + d*x)*sqrt(a + b*x)*b**4*d**4*x**3 - 15*sqrt(d)*sqrt(b)*log((sqrt( d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**4*d**4 + 60* sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a *d - b*c))*a**3*b*c*d**3 - 90*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**2*b**2*c**2*d**2 + 60*sqrt(d)* sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c ))*a*b**3*c**3*d - 15*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b) *sqrt(c + d*x))/sqrt(a*d - b*c))*b**4*c**4)/(192*b**4*d**2)