Optimal. Leaf size=66 \[ \frac {4 d \sqrt {c+d x}}{3 \sqrt {a+b x} (b c-a d)^2}-\frac {2 \sqrt {c+d x}}{3 (a+b x)^{3/2} (b c-a d)} \]
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Rubi [A] time = 0.01, antiderivative size = 66, 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} \begin {gather*} \frac {4 d \sqrt {c+d x}}{3 \sqrt {a+b x} (b c-a d)^2}-\frac {2 \sqrt {c+d x}}{3 (a+b x)^{3/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{5/2} \sqrt {c+d x}} \, dx &=-\frac {2 \sqrt {c+d x}}{3 (b c-a d) (a+b x)^{3/2}}-\frac {(2 d) \int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx}{3 (b c-a d)}\\ &=-\frac {2 \sqrt {c+d x}}{3 (b c-a d) (a+b x)^{3/2}}+\frac {4 d \sqrt {c+d x}}{3 (b c-a d)^2 \sqrt {a+b x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 46, normalized size = 0.70 \begin {gather*} \frac {2 \sqrt {c+d x} (3 a d-b c+2 b d x)}{3 (a+b x)^{3/2} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 56, normalized size = 0.85 \begin {gather*} -\frac {2 \left (\frac {b (c+d x)^{3/2}}{(a+b x)^{3/2}}-\frac {3 d \sqrt {c+d x}}{\sqrt {a+b x}}\right )}{3 (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.92, size = 118, normalized size = 1.79 \begin {gather*} \frac {2 \, {\left (2 \, b d x - b c + 3 \, a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.08, size = 121, normalized size = 1.83 \begin {gather*} \frac {8 \, {\left (b^{2} c - a b d - 3 \, {\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 )} \sqrt {b d} b^{2} d}{3 \, {\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 )}^{3} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 54, normalized size = 0.82 \begin {gather*} \frac {2 \sqrt {d x +c}\, \left (2 b d x +3 a d -b c \right )}{3 \left (b x +a \right )^{\frac {3}{2}} \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 = 0.89, size = 71, normalized size = 1.08 \begin {gather*} \frac {\left (\frac {4\,d\,x}{3\,{\left (a\,d-b\,c\right )}^2}+\frac {6\,a\,d-2\,b\,c}{3\,b\,{\left (a\,d-b\,c\right )}^2}\right )\,\sqrt {c+d\,x}}{x\,\sqrt {a+b\,x}+\frac {a\,\sqrt {a+b\,x}}{b}} \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 {5}{2}} \sqrt {c + d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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