Optimal. Leaf size=163 \[ \frac {(b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 a^2 c x}-\frac {(b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 a c x^2}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 c x^3}-\frac {(b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{5/2} c^{3/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 163, 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 {(b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{5/2} c^{3/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-a d)^2}{8 a^2 c x}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 c x^3}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}{12 a c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 96
Rule 214
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^4} \, dx &=-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 c x^3}+\frac {(b c-a d) \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx}{6 c}\\ &=-\frac {(b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 a c x^2}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 c x^3}-\frac {(b c-a d)^2 \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx}{8 a c}\\ &=\frac {(b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 a^2 c x}-\frac {(b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 a c x^2}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 c x^3}+\frac {(b c-a d)^3 \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 a^2 c}\\ &=\frac {(b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 a^2 c x}-\frac {(b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 a c x^2}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 c x^3}+\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{8 a^2 c}\\ &=\frac {(b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 a^2 c x}-\frac {(b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 a c x^2}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 c x^3}-\frac {(b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{5/2} c^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 146, normalized size = 0.90 \begin {gather*} \frac {(-b c+a d)^3 \left (\frac {\sqrt {a} \sqrt {c} \sqrt {a+b x} \sqrt {c+d x} \left (-3 b^2 c^2 x^2+2 a b c x (c+4 d x)+a^2 \left (8 c^2+14 c d x+3 d^2 x^2\right )\right )}{(b c-a d)^3 x^3}+3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )}{24 a^{5/2} c^{3/2}} \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. \(407\) vs.
\(2(131)=262\).
time = 0.07, size = 408, normalized size = 2.50
method | result | size |
default | \(\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} d^{3} x^{3}-9 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b c \,d^{2} x^{3}+9 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} d \,x^{3}-3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{3} c^{3} x^{3}-6 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} d^{2} x^{2}-16 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b c d \,x^{2}+6 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{2} c^{2} x^{2}-28 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} c d x -4 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b \,c^{2} x -16 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} c^{2} \sqrt {a c}\right )}{48 a^{2} c \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{3} \sqrt {a c}}\) | \(408\) |
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 [A]
time = 1.07, size = 438, normalized size = 2.69 \begin {gather*} \left [-\frac {3 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a c} x^{3} \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 + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (8 \, a^{3} c^{3} - {\left (3 \, a b^{2} c^{3} - 8 \, a^{2} b c^{2} d - 3 \, a^{3} c d^{2}\right )} x^{2} + 2 \, {\left (a^{2} b c^{3} + 7 \, a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, a^{3} c^{2} x^{3}}, \frac {3 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-a c} x^{3} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (8 \, a^{3} c^{3} - {\left (3 \, a b^{2} c^{3} - 8 \, a^{2} b c^{2} d - 3 \, a^{3} c d^{2}\right )} x^{2} + 2 \, {\left (a^{2} b c^{3} + 7 \, a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, a^{3} c^{2} x^{3}}\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 {\sqrt {a + b x} \left (c + d x\right )^{\frac {3}{2}}}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2161 vs.
\(2 (131) = 262\).
time = 7.17, size = 2161, normalized size = 13.26 \begin {gather*} \text {Too large to display} \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 {\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{3/2}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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