Optimal. Leaf size=75 \[ \frac {2 \sqrt {c+d x} (a d+b c)}{d \sqrt {a+b x} (b c-a d)^2}-\frac {2 c}{d \sqrt {a+b x} \sqrt {c+d x} (b c-a d)} \]
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Rubi [A] time = 0.02, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {78, 37} \begin {gather*} \frac {2 \sqrt {c+d x} (a d+b c)}{d \sqrt {a+b x} (b c-a d)^2}-\frac {2 c}{d \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 78
Rubi steps
\begin {align*} \int \frac {x}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx &=-\frac {2 c}{d (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {(b c+a d) \int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx}{d (b c-a d)}\\ &=-\frac {2 c}{d (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}+\frac {2 (b c+a d) \sqrt {c+d x}}{d (b c-a d)^2 \sqrt {a+b x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 0.57 \begin {gather*} \frac {2 (2 a c+a d x+b c x)}{\sqrt {a+b x} \sqrt {c+d x} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 46, normalized size = 0.61 \begin {gather*} \frac {2 \sqrt {a+b x} \left (\frac {a (c+d x)}{a+b x}+c\right )}{\sqrt {c+d x} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.15, size = 127, normalized size = 1.69 \begin {gather*} \frac {2 \, {\left (2 \, a c + {\left (b c + a d\right )} x\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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giac [B] time = 1.73, size = 147, normalized size = 1.96 \begin {gather*} \frac {2 \, {\left (\frac {\sqrt {b x + a} b^{3} c}{{\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 {2 \, \sqrt {b d} a 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 )}}\right )}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 53, normalized size = 0.71 \begin {gather*} \frac {2 a d x +2 b c x +4 a c}{\sqrt {b x +a}\, \sqrt {d x +c}\, \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )} \end {gather*}
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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mupad [B] time = 1.41, size = 74, normalized size = 0.99 \begin {gather*} \frac {\left (\frac {x\,\left (2\,a\,d+2\,b\,c\right )}{d\,{\left (a\,d-b\,c\right )}^2}+\frac {4\,a\,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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\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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