Optimal. Leaf size=96 \[ -\frac {6 b^2 \sqrt {d+e x} (b d-a e)}{e^4}-\frac {6 b (b d-a e)^2}{e^4 \sqrt {d+e x}}+\frac {2 (b d-a e)^3}{3 e^4 (d+e x)^{3/2}}+\frac {2 b^3 (d+e x)^{3/2}}{3 e^4} \]
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Rubi [A] time = 0.03, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 43} \[ -\frac {6 b^2 \sqrt {d+e x} (b d-a e)}{e^4}-\frac {6 b (b d-a e)^2}{e^4 \sqrt {d+e x}}+\frac {2 (b d-a e)^3}{3 e^4 (d+e x)^{3/2}}+\frac {2 b^3 (d+e x)^{3/2}}{3 e^4} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^{5/2}} \, dx &=\int \frac {(a+b x)^3}{(d+e x)^{5/2}} \, dx\\ &=\int \left (\frac {(-b d+a e)^3}{e^3 (d+e x)^{5/2}}+\frac {3 b (b d-a e)^2}{e^3 (d+e x)^{3/2}}-\frac {3 b^2 (b d-a e)}{e^3 \sqrt {d+e x}}+\frac {b^3 \sqrt {d+e x}}{e^3}\right ) \, dx\\ &=\frac {2 (b d-a e)^3}{3 e^4 (d+e x)^{3/2}}-\frac {6 b (b d-a e)^2}{e^4 \sqrt {d+e x}}-\frac {6 b^2 (b d-a e) \sqrt {d+e x}}{e^4}+\frac {2 b^3 (d+e x)^{3/2}}{3 e^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 76, normalized size = 0.79 \[ \frac {2 \left (-9 b^2 (d+e x)^2 (b d-a e)-9 b (d+e x) (b d-a e)^2+(b d-a e)^3+b^3 (d+e x)^3\right )}{3 e^4 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 136, normalized size = 1.42 \[ \frac {2 \, {\left (b^{3} e^{3} x^{3} - 16 \, b^{3} d^{3} + 24 \, a b^{2} d^{2} e - 6 \, a^{2} b d e^{2} - a^{3} e^{3} - 3 \, {\left (2 \, b^{3} d e^{2} - 3 \, a b^{2} e^{3}\right )} x^{2} - 3 \, {\left (8 \, b^{3} d^{2} e - 12 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 142, normalized size = 1.48 \[ \frac {2}{3} \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} b^{3} e^{8} - 9 \, \sqrt {x e + d} b^{3} d e^{8} + 9 \, \sqrt {x e + d} a b^{2} e^{9}\right )} e^{\left (-12\right )} - \frac {2 \, {\left (9 \, {\left (x e + d\right )} b^{3} d^{2} - b^{3} d^{3} - 18 \, {\left (x e + d\right )} a b^{2} d e + 3 \, a b^{2} d^{2} e + 9 \, {\left (x e + d\right )} a^{2} b e^{2} - 3 \, a^{2} b d e^{2} + a^{3} e^{3}\right )} e^{\left (-4\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 115, normalized size = 1.20 \[ -\frac {2 \left (-b^{3} e^{3} x^{3}-9 a \,b^{2} e^{3} x^{2}+6 b^{3} d \,e^{2} x^{2}+9 a^{2} b \,e^{3} x -36 a \,b^{2} d \,e^{2} x +24 b^{3} d^{2} e x +a^{3} e^{3}+6 a^{2} b d \,e^{2}-24 a \,b^{2} d^{2} e +16 b^{3} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 122, normalized size = 1.27 \[ \frac {2 \, {\left (\frac {{\left (e x + d\right )}^{\frac {3}{2}} b^{3} - 9 \, {\left (b^{3} d - a b^{2} e\right )} \sqrt {e x + d}}{e^{3}} + \frac {b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3} - 9 \, {\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{3}}\right )}}{3 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 128, normalized size = 1.33 \[ \frac {2\,b^3\,{\left (d+e\,x\right )}^3-2\,a^3\,e^3+2\,b^3\,d^3-18\,b^3\,d\,{\left (d+e\,x\right )}^2-18\,b^3\,d^2\,\left (d+e\,x\right )+18\,a\,b^2\,e\,{\left (d+e\,x\right )}^2-18\,a^2\,b\,e^2\,\left (d+e\,x\right )-6\,a\,b^2\,d^2\,e+6\,a^2\,b\,d\,e^2+36\,a\,b^2\,d\,e\,\left (d+e\,x\right )}{3\,e^4\,{\left (d+e\,x\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.55, size = 461, normalized size = 4.80 \[ \begin {cases} - \frac {2 a^{3} e^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {12 a^{2} b d e^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {18 a^{2} b e^{3} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {48 a b^{2} d^{2} e}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {72 a b^{2} d e^{2} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {18 a b^{2} e^{3} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {32 b^{3} d^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {48 b^{3} d^{2} e x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {12 b^{3} d e^{2} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {2 b^{3} e^{3} x^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {a^{3} x + \frac {3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac {b^{3} x^{4}}{4}}{d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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