Integrand size = 19, antiderivative size = 120 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{7/2}} \, dx=-\frac {2 d^2 \sqrt {c+d x}}{b^3 \sqrt {a+b x}}-\frac {2 d (c+d x)^{3/2}}{3 b^2 (a+b x)^{3/2}}-\frac {2 (c+d x)^{5/2}}{5 b (a+b x)^{5/2}}+\frac {2 d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{7/2}} \]
-2/3*d*(d*x+c)^(3/2)/b^2/(b*x+a)^(3/2)-2/5*(d*x+c)^(5/2)/b/(b*x+a)^(5/2)+2 *d^(5/2)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(7/2)-2*d^ 2*(d*x+c)^(1/2)/b^3/(b*x+a)^(1/2)
Time = 0.15 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.92 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{7/2}} \, dx=-\frac {2 \sqrt {c+d x} \left (15 a^2 d^2+5 a b d (c+7 d x)+b^2 \left (3 c^2+11 c d x+23 d^2 x^2\right )\right )}{15 b^3 (a+b x)^{5/2}}+\frac {2 d^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{b^{7/2}} \]
(-2*Sqrt[c + d*x]*(15*a^2*d^2 + 5*a*b*d*(c + 7*d*x) + b^2*(3*c^2 + 11*c*d* x + 23*d^2*x^2)))/(15*b^3*(a + b*x)^(5/2)) + (2*d^(5/2)*ArcTanh[(Sqrt[b]*S qrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/b^(7/2)
Time = 0.20 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {57, 57, 57, 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.
| \(\displaystyle \int \frac {(c+d x)^{5/2}}{(a+b x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle \frac {d \int \frac {(c+d x)^{3/2}}{(a+b x)^{5/2}}dx}{b}-\frac {2 (c+d x)^{5/2}}{5 b (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle \frac {d \left (\frac {d \int \frac {\sqrt {c+d x}}{(a+b x)^{3/2}}dx}{b}-\frac {2 (c+d x)^{3/2}}{3 b (a+b x)^{3/2}}\right )}{b}-\frac {2 (c+d x)^{5/2}}{5 b (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {d \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{b}-\frac {2 \sqrt {c+d x}}{b \sqrt {a+b x}}\right )}{b}-\frac {2 (c+d x)^{3/2}}{3 b (a+b x)^{3/2}}\right )}{b}-\frac {2 (c+d x)^{5/2}}{5 b (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {2 d \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{b}-\frac {2 \sqrt {c+d x}}{b \sqrt {a+b x}}\right )}{b}-\frac {2 (c+d x)^{3/2}}{3 b (a+b x)^{3/2}}\right )}{b}-\frac {2 (c+d x)^{5/2}}{5 b (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2}}-\frac {2 \sqrt {c+d x}}{b \sqrt {a+b x}}\right )}{b}-\frac {2 (c+d x)^{3/2}}{3 b (a+b x)^{3/2}}\right )}{b}-\frac {2 (c+d x)^{5/2}}{5 b (a+b x)^{5/2}}\) |
(-2*(c + d*x)^(5/2))/(5*b*(a + b*x)^(5/2)) + (d*((-2*(c + d*x)^(3/2))/(3*b *(a + b*x)^(3/2)) + (d*((-2*Sqrt[c + d*x])/(b*Sqrt[a + b*x]) + (2*Sqrt[d]* ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/b^(3/2)))/b))/b
3.15.87.3.1 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 + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[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]
\[\int \frac {\left (d x +c \right )^{\frac {5}{2}}}{\left (b x +a \right )^{\frac {7}{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (92) = 184\).
Time = 0.49 (sec) , antiderivative size = 463, normalized size of antiderivative = 3.86 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{7/2}} \, dx=\left [\frac {15 \, {\left (b^{3} d^{2} x^{3} + 3 \, a b^{2} d^{2} x^{2} + 3 \, a^{2} b d^{2} x + a^{3} d^{2}\right )} \sqrt {\frac {d}{b}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (23 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 5 \, a b c d + 15 \, a^{2} d^{2} + {\left (11 \, b^{2} c d + 35 \, a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{30 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}}, -\frac {15 \, {\left (b^{3} d^{2} x^{3} + 3 \, a b^{2} d^{2} x^{2} + 3 \, a^{2} b d^{2} x + a^{3} d^{2}\right )} \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right ) + 2 \, {\left (23 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 5 \, a b c d + 15 \, a^{2} d^{2} + {\left (11 \, b^{2} c d + 35 \, a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{15 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}}\right ] \]
[1/30*(15*(b^3*d^2*x^3 + 3*a*b^2*d^2*x^2 + 3*a^2*b*d^2*x + a^3*d^2)*sqrt(d /b)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b^2*d*x + b^2 *c + a*b*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(d/b) + 8*(b^2*c*d + a*b*d^2)* x) - 4*(23*b^2*d^2*x^2 + 3*b^2*c^2 + 5*a*b*c*d + 15*a^2*d^2 + (11*b^2*c*d + 35*a*b*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^6*x^3 + 3*a*b^5*x^2 + 3*a ^2*b^4*x + a^3*b^3), -1/15*(15*(b^3*d^2*x^3 + 3*a*b^2*d^2*x^2 + 3*a^2*b*d^ 2*x + a^3*d^2)*sqrt(-d/b)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)*s qrt(d*x + c)*sqrt(-d/b)/(b*d^2*x^2 + a*c*d + (b*c*d + a*d^2)*x)) + 2*(23*b ^2*d^2*x^2 + 3*b^2*c^2 + 5*a*b*c*d + 15*a^2*d^2 + (11*b^2*c*d + 35*a*b*d^2 )*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^6*x^3 + 3*a*b^5*x^2 + 3*a^2*b^4*x + a ^3*b^3)]
\[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{7/2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{2}}}{\left (a + b x\right )^{\frac {7}{2}}}\, dx \]
Exception generated. \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{7/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 1025 vs. \(2 (92) = 184\).
Time = 0.57 (sec) , antiderivative size = 1025, normalized size of antiderivative = 8.54 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{7/2}} \, dx=-\frac {\sqrt {b d} d^{2} {\left | b \right |} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{b^{5}} - \frac {4 \, {\left (23 \, \sqrt {b d} b^{9} c^{5} d^{2} {\left | b \right |} - 115 \, \sqrt {b d} a b^{8} c^{4} d^{3} {\left | b \right |} + 230 \, \sqrt {b d} a^{2} b^{7} c^{3} d^{4} {\left | b \right |} - 230 \, \sqrt {b d} a^{3} b^{6} c^{2} d^{5} {\left | b \right |} + 115 \, \sqrt {b d} a^{4} b^{5} c d^{6} {\left | b \right |} - 23 \, \sqrt {b d} a^{5} b^{4} d^{7} {\left | b \right |} - 70 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{7} c^{4} d^{2} {\left | b \right |} + 280 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{6} c^{3} d^{3} {\left | b \right |} - 420 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{5} c^{2} d^{4} {\left | b \right |} + 280 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{4} c d^{5} {\left | b \right |} - 70 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{4} b^{3} d^{6} {\left | b \right |} + 140 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{5} c^{3} d^{2} {\left | b \right |} - 420 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{4} c^{2} d^{3} {\left | b \right |} + 420 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{3} c d^{4} {\left | b \right |} - 140 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{3} b^{2} d^{5} {\left | b \right |} - 90 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} b^{3} c^{2} d^{2} {\left | b \right |} + 180 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a b^{2} c d^{3} {\left | b \right |} - 90 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a^{2} b d^{4} {\left | b \right |} + 45 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{8} b c d^{2} {\left | b \right |} - 45 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{8} a d^{3} {\left | b \right |}\right )}}{15 \, {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{5} b^{4}} \]
-sqrt(b*d)*d^2*abs(b)*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a )*b*d - a*b*d))^2)/b^5 - 4/15*(23*sqrt(b*d)*b^9*c^5*d^2*abs(b) - 115*sqrt( b*d)*a*b^8*c^4*d^3*abs(b) + 230*sqrt(b*d)*a^2*b^7*c^3*d^4*abs(b) - 230*sqr t(b*d)*a^3*b^6*c^2*d^5*abs(b) + 115*sqrt(b*d)*a^4*b^5*c*d^6*abs(b) - 23*sq rt(b*d)*a^5*b^4*d^7*abs(b) - 70*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt( b^2*c + (b*x + a)*b*d - a*b*d))^2*b^7*c^4*d^2*abs(b) + 280*sqrt(b*d)*(sqrt (b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^6*c^3*d^3 *abs(b) - 420*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)* b*d - a*b*d))^2*a^2*b^5*c^2*d^4*abs(b) + 280*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^4*c*d^5*abs(b) - 70*s qrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2 *a^4*b^3*d^6*abs(b) + 140*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^5*c^3*d^2*abs(b) - 420*sqrt(b*d)*(sqrt(b*d)* sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^4*c^2*d^3*abs(b ) + 420*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*b^3*c*d^4*abs(b) - 140*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^3*b^2*d^5*abs(b) - 90*sqrt(b*d)*( sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^3*c^2*d ^2*abs(b) + 180*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a )*b*d - a*b*d))^6*a*b^2*c*d^3*abs(b) - 90*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x...
Timed out. \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{7/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/2}}{{\left (a+b\,x\right )}^{7/2}} \,d x \]