Optimal. Leaf size=164 \[ -\frac {(2 b c-3 a d) (b c-a d) \sqrt {a+b x}}{c^2 d \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{c x \sqrt {c+d x}}-\frac {a^{3/2} (5 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{5/2}}+\frac {2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.11, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {100, 155, 163,
65, 223, 212, 95, 214} \begin {gather*} -\frac {a^{3/2} (5 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{5/2}}+\frac {2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{3/2}}-\frac {\sqrt {a+b x} (2 b c-3 a d) (b c-a d)}{c^2 d \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{c x \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 95
Rule 100
Rule 155
Rule 163
Rule 212
Rule 214
Rule 223
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2}}{x^2 (c+d x)^{3/2}} \, dx &=-\frac {a (a+b x)^{3/2}}{c x \sqrt {c+d x}}-\frac {\int \frac {\sqrt {a+b x} \left (-\frac {1}{2} a (5 b c-3 a d)-b^2 c x\right )}{x (c+d x)^{3/2}} \, dx}{c}\\ &=-\frac {(2 b c-3 a d) (b c-a d) \sqrt {a+b x}}{c^2 d \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{c x \sqrt {c+d x}}+\frac {2 \int \frac {\frac {1}{4} a^2 d (5 b c-3 a d)+\frac {1}{2} b^3 c^2 x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{c^2 d}\\ &=-\frac {(2 b c-3 a d) (b c-a d) \sqrt {a+b x}}{c^2 d \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{c x \sqrt {c+d x}}+\frac {b^3 \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{d}+\frac {\left (a^2 (5 b c-3 a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 c^2}\\ &=-\frac {(2 b c-3 a d) (b c-a d) \sqrt {a+b x}}{c^2 d \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{c x \sqrt {c+d x}}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{d}+\frac {\left (a^2 (5 b c-3 a d)\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{c^2}\\ &=-\frac {(2 b c-3 a d) (b c-a d) \sqrt {a+b x}}{c^2 d \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{c x \sqrt {c+d x}}-\frac {a^{3/2} (5 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{5/2}}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{d}\\ &=-\frac {(2 b c-3 a d) (b c-a d) \sqrt {a+b x}}{c^2 d \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{c x \sqrt {c+d x}}-\frac {a^{3/2} (5 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{5/2}}+\frac {2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.40, size = 150, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {a+b x} \left (2 b^2 c^2 x-4 a b c d x+a^2 d (c+3 d x)\right )}{c^2 d x \sqrt {c+d x}}+\frac {a^{3/2} (-5 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{5/2}}+\frac {2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(501\) vs.
\(2(132)=264\).
time = 0.07, size = 502, normalized size = 3.06
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} d^{3} x^{2} \sqrt {b d}-5 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b c \,d^{2} x^{2} \sqrt {b d}+2 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{2} d \,x^{2} \sqrt {a c}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) \sqrt {b d}\, a^{3} c \,d^{2} x -5 \sqrt {b d}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b \,c^{2} d x +2 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, b^{3} c^{3} x -6 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} d^{2} x +8 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b c d x -4 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{2} c^{2} x -2 a^{2} c d \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}\right )}{2 c^{2} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}\, x \sqrt {a c}\, \sqrt {d x +c}\, d}\) | \(502\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 281 vs.
\(2 (132) = 264\).
time = 2.40, size = 1233, normalized size = 7.52 \begin {gather*} \left [\frac {2 \, {\left (b^{2} c^{2} d x^{2} + b^{2} c^{3} x\right )} \sqrt {\frac {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^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - {\left ({\left (5 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{2} + {\left (5 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x\right )} \sqrt {\frac {a}{c}} \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 c^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (a^{2} c d + {\left (2 \, b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, {\left (c^{2} d^{2} x^{2} + c^{3} d x\right )}}, -\frac {4 \, {\left (b^{2} c^{2} d x^{2} + b^{2} c^{3} x\right )} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) + {\left ({\left (5 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{2} + {\left (5 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x\right )} \sqrt {\frac {a}{c}} \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 c^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (a^{2} c d + {\left (2 \, b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, {\left (c^{2} d^{2} x^{2} + c^{3} d x\right )}}, \frac {{\left ({\left (5 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{2} + {\left (5 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x\right )} \sqrt {-\frac {a}{c}} \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 {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) + {\left (b^{2} c^{2} d x^{2} + b^{2} c^{3} x\right )} \sqrt {\frac {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^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 2 \, {\left (a^{2} c d + {\left (2 \, b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (c^{2} d^{2} x^{2} + c^{3} d x\right )}}, \frac {{\left ({\left (5 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{2} + {\left (5 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x\right )} \sqrt {-\frac {a}{c}} \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 {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) - 2 \, {\left (b^{2} c^{2} d x^{2} + b^{2} c^{3} x\right )} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) - 2 \, {\left (a^{2} c d + {\left (2 \, b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (c^{2} d^{2} x^{2} + c^{3} d x\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. 560 vs.
\(2 (132) = 264\).
time = 2.31, size = 560, normalized size = 3.41 \begin {gather*} -\frac {\sqrt {b d} b^{3} \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 )}{d^{2} {\left | b \right |}} - \frac {{\left (5 \, \sqrt {b d} a^{2} b^{3} c - 3 \, \sqrt {b d} a^{3} b^{2} d\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 )}{\sqrt {-a b c d} b c^{2} {\left | b \right |}} - \frac {2 \, {\left (\sqrt {b d} a^{2} b^{5} c^{2} - 2 \, \sqrt {b d} a^{3} b^{4} c d + \sqrt {b d} a^{4} b^{3} d^{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^{3} c - \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^{2} d\right )}}{{\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 )} c^{2} {\left | b \right |}} - \frac {2 \, {\left (b^{4} c^{2} {\left | b \right |} - 2 \, a b^{3} c d {\left | b \right |} + a^{2} b^{2} d^{2} {\left | b \right |}\right )} \sqrt {b x + a}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} b^{2} c^{2} d} \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 (a+b\,x\right )}^{5/2}}{x^2\,{\left (c+d\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________