Optimal. Leaf size=162 \[ -\frac {a^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}-\frac {b^{3/2} (b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{3/2}}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{c x}+\frac {b \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{c d} \]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 162, 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 = {98, 154, 157, 63, 217, 206, 93, 208} \begin {gather*} -\frac {a^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}-\frac {b^{3/2} (b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{3/2}}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{c x}+\frac {b \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{c d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 93
Rule 98
Rule 154
Rule 157
Rule 206
Rule 208
Rule 217
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2}}{x^2 \sqrt {c+d x}} \, dx &=-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{c x}-\frac {\int \frac {\sqrt {a+b x} \left (-\frac {1}{2} a (5 b c-a d)-b (b c+a d) x\right )}{x \sqrt {c+d x}} \, dx}{c}\\ &=\frac {b (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{c d}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{c x}-\frac {\int \frac {-\frac {1}{2} a^2 d (5 b c-a d)+\frac {1}{2} b^2 c (b c-5 a d) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{c d}\\ &=\frac {b (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{c d}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{c x}-\frac {\left (b^2 (b c-5 a d)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 d}+\frac {\left (a^2 (5 b c-a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 c}\\ &=\frac {b (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{c d}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{c x}-\frac {(b (b c-5 a d)) \operatorname {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-a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{c}\\ &=\frac {b (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{c d}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{c x}-\frac {a^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}-\frac {(b (b c-5 a d)) \operatorname {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 {b (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{c d}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{c x}-\frac {a^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}-\frac {b^{3/2} (b c-5 a d) \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 = 0.87, size = 176, normalized size = 1.09 \begin {gather*} \frac {a^{3/2} (a d-5 b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (b^2 c x-a^2 d\right )}{c d x}-\frac {(b c-5 a d) (b c-a d)^{3/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{3/2} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{d^{3/2} (c+d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.44, size = 233, normalized size = 1.44 \begin {gather*} \frac {\left (a^{5/2} d-5 a^{3/2} b c\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{c^{3/2}}-\frac {\sqrt {c+d x} (b c-a d) \left (-\frac {a^2 b d (c+d x)}{a+b x}+a^2 d^2-\frac {a b^2 c (c+d x)}{a+b x}+b^2 c^2\right )}{c d \sqrt {a+b x} \left (c-\frac {a (c+d x)}{a+b x}\right ) \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {\left (5 a b^{3/2} d-b^{5/2} c\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{d^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 6.64, size = 993, normalized size = 6.13 \begin {gather*} \left [-\frac {{\left (b^{2} c^{2} - 5 \, a b c d\right )} x \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 (5 \, a b c d - a^{2} d^{2}\right )} x \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 (b^{2} c x - a^{2} d\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, c d x}, \frac {2 \, {\left (b^{2} c^{2} - 5 \, a b c d\right )} x \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 (5 \, a b c d - a^{2} d^{2}\right )} x \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 (b^{2} c x - a^{2} d\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, c d x}, \frac {2 \, {\left (5 \, a b c d - a^{2} d^{2}\right )} x \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} - 5 \, a b c d\right )} x \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 ) + 4 \, {\left (b^{2} c x - a^{2} d\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, c d x}, \frac {{\left (5 \, a b c d - a^{2} d^{2}\right )} x \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} - 5 \, a b c d\right )} x \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 (b^{2} c x - a^{2} d\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, c d x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 2.51, size = 526, normalized size = 3.25 \begin {gather*} \frac {b {\left (\frac {2 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} b}{d} + \frac {{\left (\sqrt {b d} b^{2} c - 5 \, \sqrt {b d} a b d\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 )}{d^{2}} - \frac {2 \, {\left (5 \, \sqrt {b d} a^{2} b^{2} c - \sqrt {b d} a^{3} b 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} - \frac {4 \, {\left (\sqrt {b d} a^{2} b^{4} c^{2} - 2 \, \sqrt {b d} a^{3} b^{3} c d + \sqrt {b d} a^{4} b^{2} 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^{2} 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 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}\right )}}{2 \, {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 320, normalized size = 1.98 \begin {gather*} \frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (\sqrt {b d}\, a^{3} d^{2} x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-5 \sqrt {b d}\, a^{2} b c d x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+5 \sqrt {a c}\, a \,b^{2} c d x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-\sqrt {a c}\, b^{3} c^{2} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{2} c x -2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} d \right )}{2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, c d x} \end {gather*}
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]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{5/2}}{x^2\,\sqrt {c+d\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right )^{\frac {5}{2}}}{x^{2} \sqrt {c + d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________