Optimal. Leaf size=246 \[ -\frac {(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x) (d+e x)^4}+\frac {4 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x) (d+e x)^3}-\frac {3 b^2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) (d+e x)^2}+\frac {4 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) (d+e x)}+\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^5 (a+b x)} \]
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Rubi [A]
time = 0.09, antiderivative size = 246, 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 {3 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)^2}+\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^5 (a+b x) (d+e x)^3}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^5 (a+b x) (d+e x)^4}+\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^5 (a+b x)}+\frac {4 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^5 (a+b x) (d+e 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)^5} \, 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)^5} \, 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)^5} \, 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 d+a e)^4}{e^4 (d+e x)^5}-\frac {4 b (b d-a e)^3}{e^4 (d+e x)^4}+\frac {6 b^2 (b d-a e)^2}{e^4 (d+e x)^3}-\frac {4 b^3 (b d-a e)}{e^4 (d+e x)^2}+\frac {b^4}{e^4 (d+e x)}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x) (d+e x)^4}+\frac {4 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x) (d+e x)^3}-\frac {3 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)^2}+\frac {4 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) (d+e x)}+\frac {b^4 \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 = 0.07, size = 144, normalized size = 0.59 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left ((b d-a e) \left (3 a^3 e^3+a^2 b e^2 (7 d+16 e x)+a b^2 e \left (13 d^2+40 d e x+36 e^2 x^2\right )+b^3 \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )+12 b^4 (d+e x)^4 \log (d+e x)\right )}{12 e^5 (a+b x) (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 276, normalized size = 1.12
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {4 b^{3} \left (a e -b d \right ) x^{3}}{e^{2}}-\frac {3 b^{2} \left (a^{2} e^{2}+2 a b d e -3 b^{2} d^{2}\right ) x^{2}}{e^{3}}-\frac {2 b \left (2 a^{3} e^{3}+3 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -11 b^{3} d^{3}\right ) x}{3 e^{4}}-\frac {3 a^{4} e^{4}+4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}+12 a \,b^{3} d^{3} e -25 b^{4} d^{4}}{12 e^{5}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{4}}+\frac {b^{4} \ln \left (e x +d \right ) \sqrt {\left (b x +a \right )^{2}}}{e^{5} \left (b x +a \right )}\) | \(208\) |
default | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (12 \ln \left (e x +d \right ) b^{4} e^{4} x^{4}+48 \ln \left (e x +d \right ) b^{4} d \,e^{3} x^{3}+72 \ln \left (e x +d \right ) b^{4} d^{2} e^{2} x^{2}-48 a \,b^{3} e^{4} x^{3}+48 b^{4} d \,e^{3} x^{3}+48 \ln \left (e x +d \right ) b^{4} d^{3} e x -36 a^{2} b^{2} e^{4} x^{2}-72 a \,b^{3} d \,e^{3} x^{2}+108 b^{4} d^{2} e^{2} x^{2}+12 \ln \left (e x +d \right ) b^{4} d^{4}-16 a^{3} b \,e^{4} x -24 a^{2} b^{2} d \,e^{3} x -48 a \,b^{3} d^{2} e^{2} x +88 b^{4} d^{3} e x -3 a^{4} e^{4}-4 a^{3} b d \,e^{3}-6 a^{2} b^{2} d^{2} e^{2}-12 a \,b^{3} d^{3} e +25 b^{4} d^{4}\right )}{12 \left (b x +a \right )^{3} e^{5} \left (e x +d \right )^{4}}\) | \(276\) |
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 = 2.04, size = 251, normalized size = 1.02 \begin {gather*} \frac {25 \, b^{4} d^{4} - {\left (48 \, a b^{3} x^{3} + 36 \, a^{2} b^{2} x^{2} + 16 \, a^{3} b x + 3 \, a^{4}\right )} e^{4} + 4 \, {\left (12 \, b^{4} d x^{3} - 18 \, a b^{3} d x^{2} - 6 \, a^{2} b^{2} d x - a^{3} b d\right )} e^{3} + 6 \, {\left (18 \, b^{4} d^{2} x^{2} - 8 \, a b^{3} d^{2} x - a^{2} b^{2} d^{2}\right )} e^{2} + 4 \, {\left (22 \, b^{4} d^{3} x - 3 \, a b^{3} d^{3}\right )} e + 12 \, {\left (b^{4} x^{4} e^{4} + 4 \, b^{4} d x^{3} e^{3} + 6 \, b^{4} d^{2} x^{2} e^{2} + 4 \, b^{4} d^{3} x e + b^{4} d^{4}\right )} \log \left (x e + d\right )}{12 \, {\left (x^{4} e^{9} + 4 \, d x^{3} e^{8} + 6 \, d^{2} x^{2} e^{7} + 4 \, d^{3} x e^{6} + d^{4} 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 )^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.74, size = 268, normalized size = 1.09 \begin {gather*} b^{4} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (48 \, {\left (b^{4} d e^{2} \mathrm {sgn}\left (b x + a\right ) - a b^{3} e^{3} \mathrm {sgn}\left (b x + a\right )\right )} x^{3} + 36 \, {\left (3 \, b^{4} d^{2} e \mathrm {sgn}\left (b x + a\right ) - 2 \, a b^{3} d e^{2} \mathrm {sgn}\left (b x + a\right ) - a^{2} b^{2} e^{3} \mathrm {sgn}\left (b x + a\right )\right )} x^{2} + 8 \, {\left (11 \, b^{4} d^{3} \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{3} d^{2} e \mathrm {sgn}\left (b x + a\right ) - 3 \, a^{2} b^{2} d e^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, a^{3} b e^{3} \mathrm {sgn}\left (b x + a\right )\right )} x + {\left (25 \, b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 12 \, 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 ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-1\right )}\right )} e^{\left (-4\right )}}{12 \, {\left (x e + d\right )}^{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 \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 )}^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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