Optimal. Leaf size=80 \[ \frac {2 \sqrt {a+b x} (b c-3 a d)}{3 d \sqrt {c+d x} (b c-a d)^2}-\frac {2 c \sqrt {a+b x}}{3 d (c+d x)^{3/2} (b c-a d)} \]
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Rubi [A] time = 0.02, antiderivative size = 80, 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} \[ \frac {2 \sqrt {a+b x} (b c-3 a d)}{3 d \sqrt {c+d x} (b c-a d)^2}-\frac {2 c \sqrt {a+b x}}{3 d (c+d x)^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx &=-\frac {2 c \sqrt {a+b x}}{3 d (b c-a d) (c+d x)^{3/2}}+\frac {(b c-3 a d) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 d (b c-a d)}\\ &=-\frac {2 c \sqrt {a+b x}}{3 d (b c-a d) (c+d x)^{3/2}}+\frac {2 (b c-3 a d) \sqrt {a+b x}}{3 d (b c-a d)^2 \sqrt {c+d x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 46, normalized size = 0.58 \[ \frac {2 \sqrt {a+b x} (-2 a c-3 a d x+b c x)}{3 (c+d x)^{3/2} (b c-a d)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 120, normalized size = 1.50 \[ -\frac {2 \, {\left (2 \, a c - {\left (b c - 3 \, a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.54, size = 147, normalized size = 1.84 \[ \frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (b^{5} c d {\left | b \right |} - 3 \, a b^{4} d^{2} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{4} c^{2} d - 2 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}} - \frac {3 \, {\left (a b^{5} c d {\left | b \right |} - a^{2} b^{4} d^{2} {\left | b \right |}\right )}}{b^{4} c^{2} d - 2 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 55, normalized size = 0.69 \[ -\frac {2 \sqrt {b x +a}\, \left (3 a d x -b c x +2 a c \right )}{3 \left (d x +c \right )^{\frac {3}{2}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )} \]
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 = 1.43, size = 129, normalized size = 1.61 \[ -\frac {\sqrt {c+d\,x}\,\left (\frac {x\,\left (6\,d\,a^2+2\,b\,c\,a\right )}{3\,d^2\,{\left (a\,d-b\,c\right )}^2}+\frac {4\,a^2\,c}{3\,d^2\,{\left (a\,d-b\,c\right )}^2}-\frac {x^2\,\left (2\,b^2\,c-6\,a\,b\,d\right )}{3\,d^2\,{\left (a\,d-b\,c\right )}^2}\right )}{x^2\,\sqrt {a+b\,x}+\frac {c^2\,\sqrt {a+b\,x}}{d^2}+\frac {2\,c\,x\,\sqrt {a+b\,x}}{d}} \]
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 \[ \int \frac {x}{\sqrt {a + b x} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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