Optimal. Leaf size=238 \[ \frac {b^4 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (a+b x)}-\frac {(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x) (d+e x)^3}+\frac {2 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) (d+e x)^2}-\frac {6 b^2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) (d+e x)}-\frac {4 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^5 (a+b x)} \]
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Rubi [A]
time = 0.10, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {784, 21, 45}
\begin {gather*} -\frac {6 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^5 (a+b x) (d+e x)}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^5 (a+b x) (d+e x)^2}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^5 (a+b x) (d+e x)^3}+\frac {b^4 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (a+b x)}-\frac {4 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^5 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 45
Rule 784
Rubi steps
\begin {align*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^4} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \left (a b+b^2 x\right )^3}{(d+e x)^4} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^4}{(d+e x)^4} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {b^4}{e^4}+\frac {(-b d+a e)^4}{e^4 (d+e x)^4}-\frac {4 b (b d-a e)^3}{e^4 (d+e x)^3}+\frac {6 b^2 (b d-a e)^2}{e^4 (d+e x)^2}-\frac {4 b^3 (b d-a e)}{e^4 (d+e x)}\right ) \, dx}{a b+b^2 x}\\ &=\frac {b^4 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (a+b x)}-\frac {(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x) (d+e x)^3}+\frac {2 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) (d+e x)^2}-\frac {6 b^2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) (d+e x)}-\frac {4 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^5 (a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 181, normalized size = 0.76 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (a^4 e^4+2 a^3 b e^3 (d+3 e x)+6 a^2 b^2 e^2 \left (d^2+3 d e x+3 e^2 x^2\right )-2 a b^3 d e \left (11 d^2+27 d e x+18 e^2 x^2\right )+b^4 \left (13 d^4+27 d^3 e x+9 d^2 e^2 x^2-9 d e^3 x^3-3 e^4 x^4\right )+12 b^3 (b d-a e) (d+e x)^3 \log (d+e x)\right )}{3 e^5 (a+b x) (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 330, normalized size = 1.39
method | result | size |
risch | \(\frac {b^{4} x \sqrt {\left (b x +a \right )^{2}}}{e^{4} \left (b x +a \right )}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (-6 e^{3} a^{2} b^{2}+12 d \,e^{2} a \,b^{3}-6 d^{2} e \,b^{4}\right ) x^{2}-2 b \left (a^{3} e^{3}+3 a^{2} b d \,e^{2}-9 a \,b^{2} d^{2} e +5 b^{3} d^{3}\right ) x -\frac {a^{4} e^{4}+2 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-22 a \,b^{3} d^{3} e +13 b^{4} d^{4}}{3 e}\right )}{\left (b x +a \right ) e^{4} \left (e x +d \right )^{3}}+\frac {4 \sqrt {\left (b x +a \right )^{2}}\, b^{3} \left (a e -b d \right ) \ln \left (e x +d \right )}{\left (b x +a \right ) e^{5}}\) | \(222\) |
default | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (12 \ln \left (e x +d \right ) a \,b^{3} e^{4} x^{3}-12 \ln \left (e x +d \right ) b^{4} d \,e^{3} x^{3}+3 b^{4} e^{4} x^{4}+36 \ln \left (e x +d \right ) a \,b^{3} d \,e^{3} x^{2}-36 \ln \left (e x +d \right ) b^{4} d^{2} e^{2} x^{2}+9 b^{4} d \,e^{3} x^{3}+36 \ln \left (e x +d \right ) a \,b^{3} d^{2} e^{2} x -36 \ln \left (e x +d \right ) b^{4} d^{3} e x -18 a^{2} b^{2} e^{4} x^{2}+36 a \,b^{3} d \,e^{3} x^{2}-9 b^{4} d^{2} e^{2} x^{2}+12 \ln \left (e x +d \right ) a \,b^{3} d^{3} e -12 \ln \left (e x +d \right ) b^{4} d^{4}-6 a^{3} b \,e^{4} x -18 a^{2} b^{2} d \,e^{3} x +54 a \,b^{3} d^{2} e^{2} x -27 b^{4} d^{3} e x -a^{4} e^{4}-2 a^{3} b d \,e^{3}-6 a^{2} b^{2} d^{2} e^{2}+22 a \,b^{3} d^{3} e -13 b^{4} d^{4}\right )}{3 \left (b x +a \right )^{3} e^{5} \left (e x +d \right )^{3}}\) | \(330\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.02, size = 275, normalized size = 1.16 \begin {gather*} -\frac {13 \, b^{4} d^{4} - {\left (3 \, b^{4} x^{4} - 18 \, a^{2} b^{2} x^{2} - 6 \, a^{3} b x - a^{4}\right )} e^{4} - {\left (9 \, b^{4} d x^{3} + 36 \, a b^{3} d x^{2} - 18 \, a^{2} b^{2} d x - 2 \, a^{3} b d\right )} e^{3} + 3 \, {\left (3 \, b^{4} d^{2} x^{2} - 18 \, a b^{3} d^{2} x + 2 \, a^{2} b^{2} d^{2}\right )} e^{2} + {\left (27 \, b^{4} d^{3} x - 22 \, a b^{3} d^{3}\right )} e + 12 \, {\left (b^{4} d^{4} - a b^{3} x^{3} e^{4} + {\left (b^{4} d x^{3} - 3 \, a b^{3} d x^{2}\right )} e^{3} + 3 \, {\left (b^{4} d^{2} x^{2} - a b^{3} d^{2} x\right )} e^{2} + {\left (3 \, b^{4} d^{3} x - a b^{3} d^{3}\right )} e\right )} \log \left (x e + d\right )}{3 \, {\left (x^{3} e^{8} + 3 \, d x^{2} e^{7} + 3 \, d^{2} x e^{6} + d^{3} e^{5}\right )}} \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 (a + b x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.90, size = 260, normalized size = 1.09 \begin {gather*} b^{4} x e^{\left (-4\right )} \mathrm {sgn}\left (b x + a\right ) - 4 \, {\left (b^{4} d \mathrm {sgn}\left (b x + a\right ) - a b^{3} e \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (13 \, b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 22 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 18 \, {\left (b^{4} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b^{3} d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{2} b^{2} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} x^{2} + 6 \, {\left (5 \, b^{4} d^{3} e \mathrm {sgn}\left (b x + a\right ) - 9 \, a b^{3} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{2} d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{3} b e^{4} \mathrm {sgn}\left (b x + a\right )\right )} x\right )} e^{\left (-5\right )}}{3 \, {\left (x e + d\right )}^{3}} \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 \frac {\left (a+b\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{{\left (d+e\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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