Optimal. Leaf size=62 \[ -\frac {4 d \sqrt {a+b x}}{\sqrt {c+d x} (b c-a d)^2}-\frac {2}{\sqrt {a+b x} \sqrt {c+d x} (b c-a d)} \]
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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 = {45, 37} \[ -\frac {4 d \sqrt {a+b x}}{\sqrt {c+d x} (b c-a d)^2}-\frac {2}{\sqrt {a+b x} \sqrt {c+d x} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
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 = 0.02, size = 42, normalized size = 0.68 \[ -\frac {2 (a d+b (c+2 d x))}{\sqrt {a+b x} \sqrt {c+d x} (b c-a d)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 125, normalized size = 2.02 \[ -\frac {2 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.22, size = 142, normalized size = 2.29 \[ -\frac {2 \, \sqrt {b x + a} b^{2} d}{{\left (b^{2} c^{2} {\left | b \right |} - 2 \, a b c d {\left | b \right |} + a^{2} d^{2} {\left | b \right |}\right )} \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} - \frac {4 \, \sqrt {b d} b^{2}}{{\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )} {\left (b c {\left | b \right |} - a d {\left | b \right |}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 52, normalized size = 0.84 \[ -\frac {2 \left (2 b d x +a d +b c \right )}{\sqrt {b x +a}\, \sqrt {d x +c}\, \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ -\frac {\left (\frac {4\,b\,x}{{\left (a\,d-b\,c\right )}^2}+\frac {2\,a\,d+2\,b\,c}{d\,{\left (a\,d-b\,c\right )}^2}\right )\,\sqrt {c+d\,x}}{x\,\sqrt {a+b\,x}+\frac {c\,\sqrt {a+b\,x}}{d}} \]
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 \[ \int \frac {1}{\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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