Optimal. Leaf size=204 \[ -\frac {6 b^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)}{e^4 (a+b x)}-\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^4 (a+b x) \sqrt {d+e x}}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^4 (a+b x) (d+e x)^{3/2}}+\frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^4 (a+b x)} \]
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Rubi [A] time = 0.07, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {646, 43} \begin {gather*} \frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^4 (a+b x)}-\frac {6 b^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)}{e^4 (a+b x)}-\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^4 (a+b x) \sqrt {d+e x}}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^4 (a+b x) (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^3}{(d+e x)^{5/2}} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^3 (b d-a e)^3}{e^3 (d+e x)^{5/2}}+\frac {3 b^4 (b d-a e)^2}{e^3 (d+e x)^{3/2}}-\frac {3 b^5 (b d-a e)}{e^3 \sqrt {d+e x}}+\frac {b^6 \sqrt {d+e x}}{e^3}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {2 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x) (d+e x)^{3/2}}-\frac {6 b (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (a+b x) \sqrt {d+e x}}-\frac {6 b^2 (b d-a e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (a+b x)}+\frac {2 b^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 119, normalized size = 0.58 \begin {gather*} -\frac {2 \sqrt {(a+b x)^2} \left (a^3 e^3+3 a^2 b e^2 (2 d+3 e x)-3 a b^2 e \left (8 d^2+12 d e x+3 e^2 x^2\right )+b^3 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )\right )}{3 e^4 (a+b x) (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 21.98, size = 158, normalized size = 0.77 \begin {gather*} \frac {2 \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (-a^3 e^3-9 a^2 b e^2 (d+e x)+3 a^2 b d e^2-3 a b^2 d^2 e+9 a b^2 e (d+e x)^2+18 a b^2 d e (d+e x)+b^3 d^3-9 b^3 d^2 (d+e x)+b^3 (d+e x)^3-9 b^3 d (d+e x)^2\right )}{3 e^3 (d+e x)^{3/2} (a e+b e x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 136, normalized size = 0.67 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 202, normalized size = 0.99 \begin {gather*} \frac {2}{3} \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} b^{3} e^{8} \mathrm {sgn}\left (b x + a\right ) - 9 \, \sqrt {x e + d} b^{3} d e^{8} \mathrm {sgn}\left (b x + a\right ) + 9 \, \sqrt {x e + d} a b^{2} e^{9} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-12\right )} - \frac {2 \, {\left (9 \, {\left (x e + d\right )} b^{3} d^{2} \mathrm {sgn}\left (b x + a\right ) - b^{3} d^{3} \mathrm {sgn}\left (b x + a\right ) - 18 \, {\left (x e + d\right )} a b^{2} d e \mathrm {sgn}\left (b x + a\right ) + 3 \, a b^{2} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 9 \, {\left (x e + d\right )} a^{2} b e^{2} \mathrm {sgn}\left (b x + a\right ) - 3 \, a^{2} b d e^{2} \mathrm {sgn}\left (b x + a\right ) + a^{3} e^{3} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-4\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 131, normalized size = 0.64 \begin {gather*} -\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 ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{3 \left (e x +d \right )^{\frac {3}{2}} \left (b x +a \right )^{3} e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 125, normalized size = 0.61 \begin {gather*} \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 )}}{3 \, {\left (e^{5} x + d e^{4}\right )} \sqrt {e x + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 184, normalized size = 0.90 \begin {gather*} -\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {2\,a^3\,e^3+12\,a^2\,b\,d\,e^2-48\,a\,b^2\,d^2\,e+32\,b^3\,d^3}{3\,b\,e^5}+\frac {2\,x\,\left (3\,a^2\,e^2-12\,a\,b\,d\,e+8\,b^2\,d^2\right )}{e^4}-\frac {2\,b^2\,x^3}{3\,e^2}-\frac {2\,b\,x^2\,\left (3\,a\,e-2\,b\,d\right )}{e^3}\right )}{x^2\,\sqrt {d+e\,x}+\frac {a\,d\,\sqrt {d+e\,x}}{b\,e}+\frac {x\,\left (3\,a\,e^5+3\,b\,d\,e^4\right )\,\sqrt {d+e\,x}}{3\,b\,e^5}} \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 (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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