Optimal. Leaf size=157 \[ \frac {2 \left (\frac {c^2}{a^2}-\frac {d^2}{b^2}\right ) \sqrt {c+d x}}{\sqrt {a+b x}}+\frac {2 (b c-a d) (c+d x)^{3/2}}{3 a b (a+b x)^{3/2}}-\frac {2 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2}}+\frac {2 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 157, 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 {2 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2}}+\frac {2 \sqrt {c+d x} \left (\frac {c^2}{a^2}-\frac {d^2}{b^2}\right )}{\sqrt {a+b x}}+\frac {2 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2}}+\frac {2 (c+d x)^{3/2} (b c-a d)}{3 a b (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 95
Rule 100
Rule 155
Rule 163
Rule 212
Rule 214
Rule 223
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/2}}{x (a+b x)^{5/2}} \, dx &=\frac {2 (b c-a d) (c+d x)^{3/2}}{3 a b (a+b x)^{3/2}}+\frac {2 \int \frac {\sqrt {c+d x} \left (\frac {3 b c^2}{2}+\frac {3}{2} a d^2 x\right )}{x (a+b x)^{3/2}} \, dx}{3 a b}\\ &=\frac {2 \left (\frac {c^2}{a^2}-\frac {d^2}{b^2}\right ) \sqrt {c+d x}}{\sqrt {a+b x}}+\frac {2 (b c-a d) (c+d x)^{3/2}}{3 a b (a+b x)^{3/2}}-\frac {4 \int \frac {-\frac {3}{4} b^2 c^3-\frac {3}{4} a^2 d^3 x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 a^2 b^2}\\ &=\frac {2 \left (\frac {c^2}{a^2}-\frac {d^2}{b^2}\right ) \sqrt {c+d x}}{\sqrt {a+b x}}+\frac {2 (b c-a d) (c+d x)^{3/2}}{3 a b (a+b x)^{3/2}}+\frac {c^3 \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{a^2}+\frac {d^3 \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{b^2}\\ &=\frac {2 \left (\frac {c^2}{a^2}-\frac {d^2}{b^2}\right ) \sqrt {c+d x}}{\sqrt {a+b x}}+\frac {2 (b c-a d) (c+d x)^{3/2}}{3 a b (a+b x)^{3/2}}+\frac {\left (2 c^3\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{a^2}+\frac {\left (2 d^3\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 )}{b^3}\\ &=\frac {2 \left (\frac {c^2}{a^2}-\frac {d^2}{b^2}\right ) \sqrt {c+d x}}{\sqrt {a+b x}}+\frac {2 (b c-a d) (c+d x)^{3/2}}{3 a b (a+b x)^{3/2}}-\frac {2 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2}}+\frac {\left (2 d^3\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 )}{b^3}\\ &=\frac {2 \left (\frac {c^2}{a^2}-\frac {d^2}{b^2}\right ) \sqrt {c+d x}}{\sqrt {a+b x}}+\frac {2 (b c-a d) (c+d x)^{3/2}}{3 a b (a+b x)^{3/2}}-\frac {2 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2}}+\frac {2 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 2.27, size = 193, normalized size = 1.23 \begin {gather*} \frac {2 \left (\frac {b (b c-a d) \sqrt {c+d x} \left (3 a^2 d+3 b^2 c x+4 a b (c+d x)\right )}{a^2 (a+b x)^{3/2}}-\frac {3 b^{5/2} c^{5/2} \sqrt {\frac {b}{d}} \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} \left (-b x+\sqrt {\frac {b}{d}} \sqrt {a+b x} \sqrt {c+d x}\right )}{\sqrt {a} \sqrt {b} \sqrt {c}}\right )}{a^{5/2}}-3 \sqrt {\frac {b}{d}} d^3 \log \left (\sqrt {a+b x}-\sqrt {\frac {b}{d}} \sqrt {c+d x}\right )\right )}{3 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(565\) vs.
\(2(123)=246\).
time = 0.09, size = 566, normalized size = 3.61
method | result | size |
default | \(-\frac {\sqrt {d x +c}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) \sqrt {b d}\, b^{4} c^{3} x^{2}-3 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{2} b^{2} d^{3} x^{2}+6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) \sqrt {b d}\, a \,b^{3} c^{3} x -6 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{3} b \,d^{3} x +3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) \sqrt {b d}\, a^{2} b^{2} c^{3}-3 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{4} d^{3}+8 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b \,d^{2} x -2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{2} c d x -6 b^{3} c^{2} x \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+6 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} d^{2}+2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b c d -8 a \,b^{2} c^{2} \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\right )}{3 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}\, \sqrt {a c}\, \left (b x +a \right )^{\frac {3}{2}} b^{2} a^{2}}\) | \(566\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 313 vs.
\(2 (123) = 246\).
time = 1.69, size = 1361, normalized size = 8.67 \begin {gather*} \left [\frac {3 \, {\left (a^{2} b^{2} d^{2} x^{2} + 2 \, a^{3} b d^{2} x + a^{4} 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 ) + 3 \, {\left (b^{4} c^{2} x^{2} + 2 \, a b^{3} c^{2} x + a^{2} b^{2} c^{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 (4 \, a b^{2} c^{2} - a^{2} b c d - 3 \, a^{3} d^{2} + {\left (3 \, b^{3} c^{2} + a b^{2} c d - 4 \, a^{2} b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (a^{2} b^{4} x^{2} + 2 \, a^{3} b^{3} x + a^{4} b^{2}\right )}}, -\frac {6 \, {\left (a^{2} b^{2} d^{2} x^{2} + 2 \, a^{3} b d^{2} x + a^{4} 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 ) - 3 \, {\left (b^{4} c^{2} x^{2} + 2 \, a b^{3} c^{2} x + a^{2} b^{2} c^{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 (4 \, a b^{2} c^{2} - a^{2} b c d - 3 \, a^{3} d^{2} + {\left (3 \, b^{3} c^{2} + a b^{2} c d - 4 \, a^{2} b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (a^{2} b^{4} x^{2} + 2 \, a^{3} b^{3} x + a^{4} b^{2}\right )}}, \frac {6 \, {\left (b^{4} c^{2} x^{2} + 2 \, a b^{3} c^{2} x + a^{2} b^{2} c^{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 ) + 3 \, {\left (a^{2} b^{2} d^{2} x^{2} + 2 \, a^{3} b d^{2} x + a^{4} 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 (4 \, a b^{2} c^{2} - a^{2} b c d - 3 \, a^{3} d^{2} + {\left (3 \, b^{3} c^{2} + a b^{2} c d - 4 \, a^{2} b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (a^{2} b^{4} x^{2} + 2 \, a^{3} b^{3} x + a^{4} b^{2}\right )}}, \frac {3 \, {\left (b^{4} c^{2} x^{2} + 2 \, a b^{3} c^{2} x + a^{2} b^{2} c^{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 ) - 3 \, {\left (a^{2} b^{2} d^{2} x^{2} + 2 \, a^{3} b d^{2} x + a^{4} 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 (4 \, a b^{2} c^{2} - a^{2} b c d - 3 \, a^{3} d^{2} + {\left (3 \, b^{3} c^{2} + a b^{2} c d - 4 \, a^{2} b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (a^{2} b^{4} x^{2} + 2 \, a^{3} b^{3} x + a^{4} b^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d x\right )^{\frac {5}{2}}}{x \left (a + b x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^{5/2}}{x\,{\left (a+b\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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