Optimal. Leaf size=219 \[ -\frac {(b d-a e)^3 (d+e x)^{1+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (1+m) (a+b x)}+\frac {3 b (b d-a e)^2 (d+e x)^{2+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (2+m) (a+b x)}-\frac {3 b^2 (b d-a e) (d+e x)^{3+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (3+m) (a+b x)}+\frac {b^3 (d+e x)^{4+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (4+m) (a+b x)} \]
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Rubi [A]
time = 0.06, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 45}
\begin {gather*} -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (d+e x)^{m+1}}{e^4 (m+1) (a+b x)}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (d+e x)^{m+2}}{e^4 (m+2) (a+b x)}-\frac {3 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (d+e x)^{m+3}}{e^4 (m+3) (a+b x)}+\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{m+4}}{e^4 (m+4) (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 660
Rubi steps
\begin {align*} \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^3 (d+e x)^m \, 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 (d+e x)^m}{e^3}+\frac {3 b^4 (b d-a e)^2 (d+e x)^{1+m}}{e^3}-\frac {3 b^5 (b d-a e) (d+e x)^{2+m}}{e^3}+\frac {b^6 (d+e x)^{3+m}}{e^3}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac {(b d-a e)^3 (d+e x)^{1+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (1+m) (a+b x)}+\frac {3 b (b d-a e)^2 (d+e x)^{2+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (2+m) (a+b x)}-\frac {3 b^2 (b d-a e) (d+e x)^{3+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (3+m) (a+b x)}+\frac {b^3 (d+e x)^{4+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (4+m) (a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 113, normalized size = 0.52 \begin {gather*} \frac {\left ((a+b x)^2\right )^{3/2} (d+e x)^{1+m} \left (-\frac {(b d-a e)^3}{1+m}+\frac {3 b (b d-a e)^2 (d+e x)}{2+m}-\frac {3 b^2 (b d-a e) (d+e x)^2}{3+m}+\frac {b^3 (d+e x)^3}{4+m}\right )}{e^4 (a+b 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. \(401\) vs.
\(2(175)=350\).
time = 0.68, size = 402, normalized size = 1.84
method | result | size |
gosper | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (e x +d \right )^{1+m} \left (b^{3} e^{3} m^{3} x^{3}+3 a \,b^{2} e^{3} m^{3} x^{2}+6 b^{3} e^{3} m^{2} x^{3}+3 a^{2} b \,e^{3} m^{3} x +21 a \,b^{2} e^{3} m^{2} x^{2}-3 b^{3} d \,e^{2} m^{2} x^{2}+11 b^{3} e^{3} m \,x^{3}+a^{3} e^{3} m^{3}+24 a^{2} b \,e^{3} m^{2} x -6 a \,b^{2} d \,e^{2} m^{2} x +42 a \,b^{2} e^{3} m \,x^{2}-9 b^{3} d \,e^{2} m \,x^{2}+6 b^{3} x^{3} e^{3}+9 a^{3} e^{3} m^{2}-3 a^{2} b d \,e^{2} m^{2}+57 a^{2} b \,e^{3} m x -30 a \,b^{2} d \,e^{2} m x +24 a \,b^{2} e^{3} x^{2}+6 b^{3} d^{2} e m x -6 b^{3} d \,e^{2} x^{2}+26 a^{3} e^{3} m -21 a^{2} b d \,e^{2} m +36 a^{2} b \,e^{3} x +6 a \,b^{2} d^{2} e m -24 a \,b^{2} d \,e^{2} x +6 b^{3} d^{2} e x +24 e^{3} a^{3}-36 a^{2} b d \,e^{2}+24 a \,b^{2} d^{2} e -6 b^{3} d^{3}\right )}{\left (b x +a \right )^{3} e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\) | \(402\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (b^{3} e^{4} m^{3} x^{4}+3 a \,b^{2} e^{4} m^{3} x^{3}+b^{3} d \,e^{3} m^{3} x^{3}+6 b^{3} e^{4} m^{2} x^{4}+3 a^{2} b \,e^{4} m^{3} x^{2}+3 a \,b^{2} d \,e^{3} m^{3} x^{2}+21 a \,b^{2} e^{4} m^{2} x^{3}+3 b^{3} d \,e^{3} m^{2} x^{3}+11 b^{3} e^{4} m \,x^{4}+a^{3} e^{4} m^{3} x +3 a^{2} b d \,e^{3} m^{3} x +24 a^{2} b \,e^{4} m^{2} x^{2}+15 a \,b^{2} d \,e^{3} m^{2} x^{2}+42 a \,b^{2} e^{4} m \,x^{3}-3 b^{3} d^{2} e^{2} m^{2} x^{2}+2 b^{3} d \,e^{3} m \,x^{3}+6 b^{3} x^{4} e^{4}+a^{3} d \,e^{3} m^{3}+9 a^{3} e^{4} m^{2} x +21 a^{2} b d \,e^{3} m^{2} x +57 a^{2} b \,e^{4} m \,x^{2}-6 a \,b^{2} d^{2} e^{2} m^{2} x +12 a \,b^{2} d \,e^{3} m \,x^{2}+24 a \,b^{2} e^{4} x^{3}-3 b^{3} d^{2} e^{2} m \,x^{2}+9 a^{3} d \,e^{3} m^{2}+26 a^{3} e^{4} m x -3 a^{2} b \,d^{2} e^{2} m^{2}+36 a^{2} b d \,e^{3} m x +36 a^{2} b \,e^{4} x^{2}-24 a \,b^{2} d^{2} e^{2} m x +6 b^{3} d^{3} e m x +26 a^{3} d \,e^{3} m +24 a^{3} e^{4} x -21 a^{2} b \,d^{2} e^{2} m +6 a \,b^{2} d^{3} e m +24 a^{3} d \,e^{3}-36 a^{2} b \,d^{2} e^{2}+24 a \,b^{2} d^{3} e -6 b^{3} d^{4}\right ) \left (e x +d \right )^{m}}{\left (b x +a \right ) \left (3+m \right ) \left (4+m \right ) \left (2+m \right ) \left (1+m \right ) e^{4}}\) | \(563\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 296, normalized size = 1.35 \begin {gather*} \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b^{3} x^{4} e^{4} + 6 \, a b^{2} d^{3} {\left (m + 4\right )} e - 6 \, b^{3} d^{4} - 3 \, {\left (m^{2} + 7 \, m + 12\right )} a^{2} b d^{2} e^{2} + {\left (m^{3} + 9 \, m^{2} + 26 \, m + 24\right )} a^{3} d e^{3} + {\left ({\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} b^{3} d e^{3} + 3 \, {\left (m^{3} + 7 \, m^{2} + 14 \, m + 8\right )} a b^{2} e^{4}\right )} x^{3} - 3 \, {\left ({\left (m^{2} + m\right )} b^{3} d^{2} e^{2} - {\left (m^{3} + 5 \, m^{2} + 4 \, m\right )} a b^{2} d e^{3} - {\left (m^{3} + 8 \, m^{2} + 19 \, m + 12\right )} a^{2} b e^{4}\right )} x^{2} + {\left (6 \, b^{3} d^{3} m e - 6 \, {\left (m^{2} + 4 \, m\right )} a b^{2} d^{2} e^{2} + 3 \, {\left (m^{3} + 7 \, m^{2} + 12 \, m\right )} a^{2} b d e^{3} + {\left (m^{3} + 9 \, m^{2} + 26 \, m + 24\right )} a^{3} e^{4}\right )} x\right )} e^{\left (m \log \left (x e + d\right ) - 4\right )}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 421 vs.
\(2 (178) = 356\).
time = 2.54, size = 421, normalized size = 1.92 \begin {gather*} -\frac {{\left (6 \, b^{3} d^{4} - {\left ({\left (b^{3} m^{3} + 6 \, b^{3} m^{2} + 11 \, b^{3} m + 6 \, b^{3}\right )} x^{4} + 3 \, {\left (a b^{2} m^{3} + 7 \, a b^{2} m^{2} + 14 \, a b^{2} m + 8 \, a b^{2}\right )} x^{3} + 3 \, {\left (a^{2} b m^{3} + 8 \, a^{2} b m^{2} + 19 \, a^{2} b m + 12 \, a^{2} b\right )} x^{2} + {\left (a^{3} m^{3} + 9 \, a^{3} m^{2} + 26 \, a^{3} m + 24 \, a^{3}\right )} x\right )} e^{4} - {\left (a^{3} d m^{3} + 9 \, a^{3} d m^{2} + 26 \, a^{3} d m + 24 \, a^{3} d + {\left (b^{3} d m^{3} + 3 \, b^{3} d m^{2} + 2 \, b^{3} d m\right )} x^{3} + 3 \, {\left (a b^{2} d m^{3} + 5 \, a b^{2} d m^{2} + 4 \, a b^{2} d m\right )} x^{2} + 3 \, {\left (a^{2} b d m^{3} + 7 \, a^{2} b d m^{2} + 12 \, a^{2} b d m\right )} x\right )} e^{3} + 3 \, {\left (a^{2} b d^{2} m^{2} + 7 \, a^{2} b d^{2} m + 12 \, a^{2} b d^{2} + {\left (b^{3} d^{2} m^{2} + b^{3} d^{2} m\right )} x^{2} + 2 \, {\left (a b^{2} d^{2} m^{2} + 4 \, a b^{2} d^{2} m\right )} x\right )} e^{2} - 6 \, {\left (b^{3} d^{3} m x + a b^{2} d^{3} m + 4 \, a b^{2} d^{3}\right )} e\right )} {\left (x e + d\right )}^{m} e^{\left (-4\right )}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \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 \left (d + e x\right )^{m} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1075 vs.
\(2 (178) = 356\).
time = 0.91, size = 1075, normalized size = 4.91 \begin {gather*} \frac {{\left (x e + d\right )}^{m} b^{3} m^{3} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + {\left (x e + d\right )}^{m} b^{3} d m^{3} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, {\left (x e + d\right )}^{m} a b^{2} m^{3} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 6 \, {\left (x e + d\right )}^{m} b^{3} m^{2} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 3 \, {\left (x e + d\right )}^{m} a b^{2} d m^{3} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, {\left (x e + d\right )}^{m} b^{3} d m^{2} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, {\left (x e + d\right )}^{m} b^{3} d^{2} m^{2} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, {\left (x e + d\right )}^{m} a^{2} b m^{3} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 21 \, {\left (x e + d\right )}^{m} a b^{2} m^{2} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 11 \, {\left (x e + d\right )}^{m} b^{3} m x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 3 \, {\left (x e + d\right )}^{m} a^{2} b d m^{3} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, {\left (x e + d\right )}^{m} a b^{2} d m^{2} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, {\left (x e + d\right )}^{m} b^{3} d m x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 6 \, {\left (x e + d\right )}^{m} a b^{2} d^{2} m^{2} x e^{2} \mathrm {sgn}\left (b x + a\right ) - 3 \, {\left (x e + d\right )}^{m} b^{3} d^{2} m x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, {\left (x e + d\right )}^{m} b^{3} d^{3} m x e \mathrm {sgn}\left (b x + a\right ) + {\left (x e + d\right )}^{m} a^{3} m^{3} x e^{4} \mathrm {sgn}\left (b x + a\right ) + 24 \, {\left (x e + d\right )}^{m} a^{2} b m^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 42 \, {\left (x e + d\right )}^{m} a b^{2} m x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 6 \, {\left (x e + d\right )}^{m} b^{3} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + {\left (x e + d\right )}^{m} a^{3} d m^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 21 \, {\left (x e + d\right )}^{m} a^{2} b d m^{2} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 12 \, {\left (x e + d\right )}^{m} a b^{2} d m x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, {\left (x e + d\right )}^{m} a^{2} b d^{2} m^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 24 \, {\left (x e + d\right )}^{m} a b^{2} d^{2} m x e^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, {\left (x e + d\right )}^{m} a b^{2} d^{3} m e \mathrm {sgn}\left (b x + a\right ) - 6 \, {\left (x e + d\right )}^{m} b^{3} d^{4} \mathrm {sgn}\left (b x + a\right ) + 9 \, {\left (x e + d\right )}^{m} a^{3} m^{2} x e^{4} \mathrm {sgn}\left (b x + a\right ) + 57 \, {\left (x e + d\right )}^{m} a^{2} b m x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 24 \, {\left (x e + d\right )}^{m} a b^{2} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 9 \, {\left (x e + d\right )}^{m} a^{3} d m^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 36 \, {\left (x e + d\right )}^{m} a^{2} b d m x e^{3} \mathrm {sgn}\left (b x + a\right ) - 21 \, {\left (x e + d\right )}^{m} a^{2} b d^{2} m e^{2} \mathrm {sgn}\left (b x + a\right ) + 24 \, {\left (x e + d\right )}^{m} a b^{2} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 26 \, {\left (x e + d\right )}^{m} a^{3} m x e^{4} \mathrm {sgn}\left (b x + a\right ) + 36 \, {\left (x e + d\right )}^{m} a^{2} b x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 26 \, {\left (x e + d\right )}^{m} a^{3} d m e^{3} \mathrm {sgn}\left (b x + a\right ) - 36 \, {\left (x e + d\right )}^{m} a^{2} b d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 24 \, {\left (x e + d\right )}^{m} a^{3} x e^{4} \mathrm {sgn}\left (b x + a\right ) + 24 \, {\left (x e + d\right )}^{m} a^{3} d e^{3} \mathrm {sgn}\left (b x + a\right )}{m^{4} e^{4} + 10 \, m^{3} e^{4} + 35 \, m^{2} e^{4} + 50 \, m e^{4} + 24 \, e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (d+e\,x\right )}^m\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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