Optimal. Leaf size=54 \[ -\frac {2 (d+e x)^3}{3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {664}
\begin {gather*} -\frac {2 (d+e x)^3}{3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 664
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^3}{3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 46, normalized size = 0.85 \begin {gather*} -\frac {2 ((a e+c d x) (d+e x))^{3/2}}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1535\) vs.
\(2(50)=100\).
time = 0.84, size = 1536, normalized size = 28.44
method | result | size |
gosper | \(\frac {2 \left (c d x +a e \right ) \left (e x +d \right )^{4}}{3 \left (e^{2} a -c \,d^{2}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\) | \(58\) |
trager | \(\frac {2 \left (e x +d \right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{3 \left (c d x +a e \right )^{2} \left (e^{2} a -c \,d^{2}\right )}\) | \(58\) |
default | \(\text {Expression too large to display}\) | \(1536\) |
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 [A]
time = 2.43, size = 102, normalized size = 1.89 \begin {gather*} -\frac {2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (x e + d\right )}}{3 \, {\left (c^{3} d^{4} x^{2} + 2 \, a c^{2} d^{3} x e - 2 \, a^{2} c d x e^{3} - a^{3} e^{4} - {\left (a c^{2} d^{2} x^{2} - a^{2} c d^{2}\right )} e^{2}\right )}} \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 {\left (d + e x\right )^{3}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.43, size = 58, normalized size = 1.07 \begin {gather*} \frac {2\,\left (d+e\,x\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{3\,{\left (a\,e+c\,d\,x\right )}^2\,\left (a\,e^2-c\,d^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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