Optimal. Leaf size=202 \[ -\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{e^4 (a+b x)}+\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^2}{e^4 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^4 (a+b x) \sqrt {d+e x}}+\frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^4 (a+b x)} \]
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Rubi [A] time = 0.06, antiderivative size = 202, 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)^{5/2}}{5 e^4 (a+b x)}-\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{e^4 (a+b x)}+\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^2}{e^4 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^4 (a+b x) \sqrt {d+e x}} \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)^{3/2}} \, 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)^{3/2}} \, 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)^{3/2}}+\frac {3 b^4 (b d-a e)^2}{e^3 \sqrt {d+e x}}-\frac {3 b^5 (b d-a e) \sqrt {d+e x}}{e^3}+\frac {b^6 (d+e x)^{3/2}}{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}}{e^4 (a+b x) \sqrt {d+e x}}+\frac {6 b (b d-a e)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (a+b x)}-\frac {2 b^2 (b d-a e) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (a+b x)}+\frac {2 b^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^4 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 119, normalized size = 0.59 \begin {gather*} -\frac {2 \sqrt {(a+b x)^2} \left (5 a^3 e^3-15 a^2 b e^2 (2 d+e x)+5 a b^2 e \left (8 d^2+4 d e x-e^2 x^2\right )-\left (b^3 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )\right )}{5 e^4 (a+b x) \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 19.36, size = 159, normalized size = 0.79 \begin {gather*} \frac {2 \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (-5 a^3 e^3+15 a^2 b e^2 (d+e x)+15 a^2 b d e^2-15 a b^2 d^2 e+5 a b^2 e (d+e x)^2-30 a b^2 d e (d+e x)+5 b^3 d^3+15 b^3 d^2 (d+e x)+b^3 (d+e x)^3-5 b^3 d (d+e x)^2\right )}{5 e^3 \sqrt {d+e x} (a e+b e x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 124, normalized size = 0.61 \begin {gather*} \frac {2 \, {\left (b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} - 40 \, a b^{2} d^{2} e + 30 \, a^{2} b d e^{2} - 5 \, a^{3} e^{3} - {\left (2 \, b^{3} d e^{2} - 5 \, a b^{2} e^{3}\right )} x^{2} + {\left (8 \, b^{3} d^{2} e - 20 \, a b^{2} d e^{2} + 15 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{5 \, {\left (e^{5} x + d e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 210, normalized size = 1.04 \begin {gather*} \frac {2}{5} \, {\left ({\left (x e + d\right )}^{\frac {5}{2}} b^{3} e^{16} \mathrm {sgn}\left (b x + a\right ) - 5 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} d e^{16} \mathrm {sgn}\left (b x + a\right ) + 15 \, \sqrt {x e + d} b^{3} d^{2} e^{16} \mathrm {sgn}\left (b x + a\right ) + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{2} e^{17} \mathrm {sgn}\left (b x + a\right ) - 30 \, \sqrt {x e + d} a b^{2} d e^{17} \mathrm {sgn}\left (b x + a\right ) + 15 \, \sqrt {x e + d} a^{2} b e^{18} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-20\right )} + \frac {2 \, {\left (b^{3} d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{2} d^{2} e \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 )}}{\sqrt {x e + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 132, normalized size = 0.65 \begin {gather*} -\frac {2 \left (-b^{3} e^{3} x^{3}-5 a \,b^{2} e^{3} x^{2}+2 b^{3} d \,e^{2} x^{2}-15 a^{2} b \,e^{3} x +20 a \,b^{2} d \,e^{2} x -8 b^{3} d^{2} e x +5 a^{3} e^{3}-30 a^{2} b d \,e^{2}+40 a \,b^{2} d^{2} e -16 b^{3} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{5 \sqrt {e x +d}\, \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 = 1.14, size = 114, normalized size = 0.56 \begin {gather*} \frac {2 \, {\left (b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} - 40 \, a b^{2} d^{2} e + 30 \, a^{2} b d e^{2} - 5 \, a^{3} e^{3} - {\left (2 \, b^{3} d e^{2} - 5 \, a b^{2} e^{3}\right )} x^{2} + {\left (8 \, b^{3} d^{2} e - 20 \, a b^{2} d e^{2} + 15 \, a^{2} b e^{3}\right )} x\right )}}{5 \, \sqrt {e x + d} e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.15, size = 147, normalized size = 0.73 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {2\,x\,\left (15\,a^2\,e^2-20\,a\,b\,d\,e+8\,b^2\,d^2\right )}{5\,e^3}-\frac {2\,a^3\,e^3-12\,a^2\,b\,d\,e^2+16\,a\,b^2\,d^2\,e-\frac {32\,b^3\,d^3}{5}}{b\,e^4}+\frac {2\,b^2\,x^3}{5\,e}+\frac {2\,b\,x^2\,\left (5\,a\,e-2\,b\,d\right )}{5\,e^2}\right )}{x\,\sqrt {d+e\,x}+\frac {a\,\sqrt {d+e\,x}}{b}} \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 {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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