Optimal. Leaf size=62 \[ -\frac {2}{(b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {4 d \sqrt {a+b x}}{(b c-a d)^2 \sqrt {c+d x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37}
\begin {gather*} -\frac {4 d \sqrt {a+b x}}{\sqrt {c+d x} (b c-a d)^2}-\frac {2}{\sqrt {a+b x} \sqrt {c+d x} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx &=-\frac {2}{(b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {(2 d) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{b c-a d}\\ &=-\frac {2}{(b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {4 d \sqrt {a+b x}}{(b c-a d)^2 \sqrt {c+d x}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 42, normalized size = 0.68 \begin {gather*} -\frac {2 (a d+b (c+2 d x))}{(b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.16, size = 65, normalized size = 1.05
method | result | size |
gosper | \(-\frac {2 \left (2 b d x +a d +b c \right )}{\sqrt {b x +a}\, \sqrt {d x +c}\, \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(52\) |
default | \(-\frac {2}{\left (-a d +b c \right ) \sqrt {b x +a}\, \sqrt {d x +c}}+\frac {4 d \sqrt {b x +a}}{\left (-a d +b c \right ) \sqrt {d x +c}\, \left (a d -b c \right )}\) | \(65\) |
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. 125 vs.
\(2 (54) = 108\).
time = 0.34, size = 125, normalized size = 2.02 \begin {gather*} -\frac {2 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x} \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 {1}{\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{2}}}\, 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. 142 vs.
\(2 (54) = 108\).
time = 0.02, size = 177, normalized size = 2.85 \begin {gather*} 2 \left (-\frac {2 b^{2} d \sqrt {a+b x} \sqrt {-a b d+b^{2} c+b d \left (a+b x\right )}}{\left (2 b^{2} c^{2} \left |b\right |-4 b d a c \left |b\right |+2 d^{2} a^{2} \left |b\right |\right ) \left (-a b d+b^{2} c+b d \left (a+b x\right )\right )}-\frac {4 b^{2} \sqrt {b d}}{2 \left (a d \left |b\right |-\left |b\right | b c\right ) \left (\left (\sqrt {-a b d+b^{2} c+b d \left (a+b x\right )}-\sqrt {b d} \sqrt {a+b x}\right )^{2}+a d b-b^{2} c\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.86, size = 71, normalized size = 1.15 \begin {gather*} -\frac {\left (\frac {4\,b\,x}{{\left (a\,d-b\,c\right )}^2}+\frac {2\,a\,d+2\,b\,c}{d\,{\left (a\,d-b\,c\right )}^2}\right )\,\sqrt {c+d\,x}}{x\,\sqrt {a+b\,x}+\frac {c\,\sqrt {a+b\,x}}{d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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