Optimal. Leaf size=154 \[ \frac {15 (b c-a d)^2 \sqrt {c+d x}}{4 a^3 \sqrt {a+b x}}+\frac {5 (b c-a d) (c+d x)^{3/2}}{4 a^2 x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a x^2 \sqrt {a+b x}}-\frac {15 \sqrt {c} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {96, 95, 214}
\begin {gather*} -\frac {15 \sqrt {c} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{7/2}}+\frac {15 \sqrt {c+d x} (b c-a d)^2}{4 a^3 \sqrt {a+b x}}+\frac {5 (c+d x)^{3/2} (b c-a d)}{4 a^2 x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a x^2 \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 95
Rule 96
Rule 214
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^{3/2}} \, dx &=-\frac {(c+d x)^{5/2}}{2 a x^2 \sqrt {a+b x}}-\frac {(5 (b c-a d)) \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^{3/2}} \, dx}{4 a}\\ &=\frac {5 (b c-a d) (c+d x)^{3/2}}{4 a^2 x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a x^2 \sqrt {a+b x}}+\frac {\left (15 (b c-a d)^2\right ) \int \frac {\sqrt {c+d x}}{x (a+b x)^{3/2}} \, dx}{8 a^2}\\ &=\frac {15 (b c-a d)^2 \sqrt {c+d x}}{4 a^3 \sqrt {a+b x}}+\frac {5 (b c-a d) (c+d x)^{3/2}}{4 a^2 x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a x^2 \sqrt {a+b x}}+\frac {\left (15 c (b c-a d)^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 a^3}\\ &=\frac {15 (b c-a d)^2 \sqrt {c+d x}}{4 a^3 \sqrt {a+b x}}+\frac {5 (b c-a d) (c+d x)^{3/2}}{4 a^2 x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a x^2 \sqrt {a+b x}}+\frac {\left (15 c (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 a^3}\\ &=\frac {15 (b c-a d)^2 \sqrt {c+d x}}{4 a^3 \sqrt {a+b x}}+\frac {5 (b c-a d) (c+d x)^{3/2}}{4 a^2 x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a x^2 \sqrt {a+b x}}-\frac {15 \sqrt {c} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.25, size = 130, normalized size = 0.84 \begin {gather*} \frac {\sqrt {c+d x} \left (15 b^2 c^2 x^2+5 a b c x (c-5 d x)+a^2 \left (-2 c^2-9 c d x+8 d^2 x^2\right )\right )}{4 a^3 x^2 \sqrt {a+b x}}-\frac {15 \sqrt {c} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{4 a^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(506\) vs.
\(2(122)=244\).
time = 0.08, size = 507, normalized size = 3.29
method | result | size |
default | \(-\frac {\sqrt {d x +c}\, \left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b c \,d^{2} x^{3}-30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} d \,x^{3}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{3} c^{3} x^{3}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} c \,d^{2} x^{2}-30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b \,c^{2} d \,x^{2}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{2} c^{3} x^{2}-16 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} d^{2} x^{2}+50 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b c d \,x^{2}-30 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{2} c^{2} x^{2}+18 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} c d x -10 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b \,c^{2} x +4 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} c^{2} \sqrt {a c}\right )}{8 a^{3} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{2} \sqrt {a c}\, \sqrt {b x +a}}\) | \(507\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.60, size = 481, normalized size = 3.12 \begin {gather*} \left [\frac {15 \, {\left ({\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}\right )} \sqrt {\frac {c}{a}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a^{2} c + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c}{a}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (2 \, a^{2} c^{2} - {\left (15 \, b^{2} c^{2} - 25 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{2} - {\left (5 \, a b c^{2} - 9 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}}, \frac {15 \, {\left ({\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}\right )} \sqrt {-\frac {c}{a}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {c}{a}}}{2 \, {\left (b c d x^{2} + a c^{2} + {\left (b c^{2} + a c d\right )} x\right )}}\right ) - 2 \, {\left (2 \, a^{2} c^{2} - {\left (15 \, b^{2} c^{2} - 25 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{2} - {\left (5 \, a b c^{2} - 9 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1206 vs.
\(2 (122) = 244\).
time = 3.93, size = 1206, normalized size = 7.83 \begin {gather*} -\frac {15 \, {\left (\sqrt {b d} b^{2} c^{3} {\left | b \right |} - 2 \, \sqrt {b d} a b c^{2} d {\left | b \right |} + \sqrt {b d} a^{2} c d^{2} {\left | b \right |}\right )} \arctan \left (-\frac {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}}{2 \, \sqrt {-a b c d} b}\right )}{4 \, \sqrt {-a b c d} a^{3} b} + \frac {7 \, \sqrt {b d} b^{8} c^{6} {\left | b \right |} - 37 \, \sqrt {b d} a b^{7} c^{5} d {\left | b \right |} + 78 \, \sqrt {b d} a^{2} b^{6} c^{4} d^{2} {\left | b \right |} - 82 \, \sqrt {b d} a^{3} b^{5} c^{3} d^{3} {\left | b \right |} + 43 \, \sqrt {b d} a^{4} b^{4} c^{2} d^{4} {\left | b \right |} - 9 \, \sqrt {b d} a^{5} b^{3} c d^{5} {\left | b \right |} - 21 \, \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^{6} c^{5} {\left | b \right |} + 44 \, \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^{5} c^{4} d {\left | b \right |} + 2 \, \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^{4} c^{3} d^{2} {\left | b \right |} - 52 \, \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^{3} c^{2} d^{3} {\left | b \right |} + 27 \, \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^{2} c d^{4} {\left | b \right |} + 21 \, \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^{4} c^{4} {\left | b \right |} - 13 \, \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^{3} c^{3} d {\left | b \right |} + 3 \, \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^{2} c^{2} d^{2} {\left | b \right |} - 27 \, \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 c d^{3} {\left | b \right |} - 7 \, \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^{2} c^{3} {\left | b \right |} + 6 \, \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 c^{2} d {\left | b \right |} + 9 \, \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} c d^{2} {\left | b \right |}}{2 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\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^{2} c - 2 \, {\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 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}\right )}^{2} a^{3}} + \frac {4 \, {\left (\sqrt {b d} b^{3} c^{3} {\left | b \right |} - 3 \, \sqrt {b d} a b^{2} c^{2} d {\left | b \right |} + 3 \, \sqrt {b d} a^{2} b c d^{2} {\left | b \right |} - \sqrt {b d} a^{3} d^{3} {\left | b \right |}\right )}}{{\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 )} a^{3} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^{5/2}}{x^3\,{\left (a+b\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________