Optimal. Leaf size=94 \[ \frac {6 b^2 (b d-a e)}{e^4 \sqrt {d+e x}}-\frac {2 b (b d-a e)^2}{e^4 (d+e x)^{3/2}}+\frac {2 (b d-a e)^3}{5 e^4 (d+e x)^{5/2}}+\frac {2 b^3 \sqrt {d+e x}}{e^4} \]
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Rubi [A] time = 0.03, antiderivative size = 94, 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} \begin {gather*} \frac {6 b^2 (b d-a e)}{e^4 \sqrt {d+e x}}-\frac {2 b (b d-a e)^2}{e^4 (d+e x)^{3/2}}+\frac {2 (b d-a e)^3}{5 e^4 (d+e x)^{5/2}}+\frac {2 b^3 \sqrt {d+e x}}{e^4} \end {gather*}
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)^{7/2}} \, dx &=\int \frac {(a+b x)^3}{(d+e x)^{7/2}} \, dx\\ &=\int \left (\frac {(-b d+a e)^3}{e^3 (d+e x)^{7/2}}+\frac {3 b (b d-a e)^2}{e^3 (d+e x)^{5/2}}-\frac {3 b^2 (b d-a e)}{e^3 (d+e x)^{3/2}}+\frac {b^3}{e^3 \sqrt {d+e x}}\right ) \, dx\\ &=\frac {2 (b d-a e)^3}{5 e^4 (d+e x)^{5/2}}-\frac {2 b (b d-a e)^2}{e^4 (d+e x)^{3/2}}+\frac {6 b^2 (b d-a e)}{e^4 \sqrt {d+e x}}+\frac {2 b^3 \sqrt {d+e x}}{e^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 77, normalized size = 0.82 \begin {gather*} \frac {2 \left (15 b^2 (d+e x)^2 (b d-a e)-5 b (d+e x) (b d-a e)^2+(b d-a e)^3+5 b^3 (d+e x)^3\right )}{5 e^4 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 131, normalized size = 1.39 \begin {gather*} \frac {2 \left (-a^3 e^3-5 a^2 b e^2 (d+e x)+3 a^2 b d e^2-3 a b^2 d^2 e-15 a b^2 e (d+e x)^2+10 a b^2 d e (d+e x)+b^3 d^3-5 b^3 d^2 (d+e x)+5 b^3 (d+e x)^3+15 b^3 d (d+e x)^2\right )}{5 e^4 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 148, normalized size = 1.57 \begin {gather*} \frac {2 \, {\left (5 \, b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} - 8 \, a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} - a^{3} e^{3} + 15 \, {\left (2 \, b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 5 \, {\left (8 \, b^{3} d^{2} e - 4 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{5 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 136, normalized size = 1.45 \begin {gather*} 2 \, \sqrt {x e + d} b^{3} e^{\left (-4\right )} + \frac {2 \, {\left (15 \, {\left (x e + d\right )}^{2} b^{3} d - 5 \, {\left (x e + d\right )} b^{3} d^{2} + b^{3} d^{3} - 15 \, {\left (x e + d\right )}^{2} a b^{2} e + 10 \, {\left (x e + d\right )} a b^{2} d e - 3 \, a b^{2} d^{2} e - 5 \, {\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 )}}{5 \, {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 115, normalized size = 1.22 \begin {gather*} -\frac {2 \left (-5 b^{3} e^{3} x^{3}+15 a \,b^{2} e^{3} x^{2}-30 b^{3} d \,e^{2} x^{2}+5 a^{2} b \,e^{3} x +20 a \,b^{2} d \,e^{2} x -40 b^{3} d^{2} e x +a^{3} e^{3}+2 a^{2} b d \,e^{2}+8 a \,b^{2} d^{2} e -16 b^{3} d^{3}\right )}{5 \left (e x +d \right )^{\frac {5}{2}} e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 121, normalized size = 1.29 \begin {gather*} \frac {2 \, {\left (\frac {5 \, \sqrt {e x + d} b^{3}}{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} + 15 \, {\left (b^{3} d - a b^{2} e\right )} {\left (e x + d\right )}^{2} - 5 \, {\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 {5}{2}} e^{3}}\right )}}{5 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.06, size = 114, normalized size = 1.21 \begin {gather*} -\frac {2\,\left (a^3\,e^3+2\,a^2\,b\,d\,e^2+5\,a^2\,b\,e^3\,x+8\,a\,b^2\,d^2\,e+20\,a\,b^2\,d\,e^2\,x+15\,a\,b^2\,e^3\,x^2-16\,b^3\,d^3-40\,b^3\,d^2\,e\,x-30\,b^3\,d\,e^2\,x^2-5\,b^3\,e^3\,x^3\right )}{5\,e^4\,{\left (d+e\,x\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.45, size = 665, normalized size = 7.07 \begin {gather*} \begin {cases} - \frac {2 a^{3} e^{3}}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} - \frac {4 a^{2} b d e^{2}}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} - \frac {10 a^{2} b e^{3} x}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} - \frac {16 a b^{2} d^{2} e}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} - \frac {40 a b^{2} d e^{2} x}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} - \frac {30 a b^{2} e^{3} x^{2}}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} + \frac {32 b^{3} d^{3}}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} + \frac {80 b^{3} d^{2} e x}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} + \frac {60 b^{3} d e^{2} x^{2}}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} + \frac {10 b^{3} e^{3} x^{3}}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \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 {7}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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