Optimal. Leaf size=298 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6 \log (d+e x)}{e^7 (a+b x)}-\frac {b x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^6 (a+b x)}+\frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^5}-\frac {(a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^4}+\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{4 e^3}-\frac {(a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^2}+\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e} \]
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Rubi [A] time = 0.16, antiderivative size = 298, 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 = {770, 21, 43} \begin {gather*} -\frac {b x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^6 (a+b x)}+\frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^5}-\frac {(a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^4}+\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{4 e^3}-\frac {(a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^2}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6 \log (d+e x)}{e^7 (a+b x)}+\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 43
Rule 770
Rubi steps
\begin {align*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, 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 )^5}{d+e x} \, dx}{b^4 \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)^6}{d+e x} \, 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 (b d-a e)^5}{e^6}+\frac {b (b d-a e)^4 (a+b x)}{e^5}-\frac {b (b d-a e)^3 (a+b x)^2}{e^4}+\frac {b (b d-a e)^2 (a+b x)^3}{e^3}-\frac {b (b d-a e) (a+b x)^4}{e^2}+\frac {b (a+b x)^5}{e}+\frac {(-b d+a e)^6}{e^6 (d+e x)}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {b (b d-a e)^5 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac {(b d-a e)^4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5}-\frac {(b d-a e)^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^4}+\frac {(b d-a e)^2 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^3}-\frac {(b d-a e) (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^2}+\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e}+\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 248, normalized size = 0.83 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (b e x \left (360 a^5 e^5+450 a^4 b e^4 (e x-2 d)+200 a^3 b^2 e^3 \left (6 d^2-3 d e x+2 e^2 x^2\right )+75 a^2 b^3 e^2 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+6 a b^4 e \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+b^5 \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )+60 (b d-a e)^6 \log (d+e x)\right )}{60 e^7 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 3.12, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{d+e x} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 351, normalized size = 1.18 \begin {gather*} \frac {10 \, b^{6} e^{6} x^{6} - 12 \, {\left (b^{6} d e^{5} - 6 \, a b^{5} e^{6}\right )} x^{5} + 15 \, {\left (b^{6} d^{2} e^{4} - 6 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \, {\left (b^{6} d^{3} e^{3} - 6 \, a b^{5} d^{2} e^{4} + 15 \, a^{2} b^{4} d e^{5} - 20 \, a^{3} b^{3} e^{6}\right )} x^{3} + 30 \, {\left (b^{6} d^{4} e^{2} - 6 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} - 20 \, a^{3} b^{3} d e^{5} + 15 \, a^{4} b^{2} e^{6}\right )} x^{2} - 60 \, {\left (b^{6} d^{5} e - 6 \, a b^{5} d^{4} e^{2} + 15 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} - 6 \, a^{5} b e^{6}\right )} x + 60 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \log \left (e x + d\right )}{60 \, e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 522, normalized size = 1.75 \begin {gather*} {\left (b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{60} \, {\left (10 \, b^{6} x^{6} e^{5} \mathrm {sgn}\left (b x + a\right ) - 12 \, b^{6} d x^{5} e^{4} \mathrm {sgn}\left (b x + a\right ) + 15 \, b^{6} d^{2} x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) - 20 \, b^{6} d^{3} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 30 \, b^{6} d^{4} x^{2} e \mathrm {sgn}\left (b x + a\right ) - 60 \, b^{6} d^{5} x \mathrm {sgn}\left (b x + a\right ) + 72 \, a b^{5} x^{5} e^{5} \mathrm {sgn}\left (b x + a\right ) - 90 \, a b^{5} d x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 120 \, a b^{5} d^{2} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 180 \, a b^{5} d^{3} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 360 \, a b^{5} d^{4} x e \mathrm {sgn}\left (b x + a\right ) + 225 \, a^{2} b^{4} x^{4} e^{5} \mathrm {sgn}\left (b x + a\right ) - 300 \, a^{2} b^{4} d x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 450 \, a^{2} b^{4} d^{2} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 900 \, a^{2} b^{4} d^{3} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 400 \, a^{3} b^{3} x^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) - 600 \, a^{3} b^{3} d x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 1200 \, a^{3} b^{3} d^{2} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 450 \, a^{4} b^{2} x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) - 900 \, a^{4} b^{2} d x e^{4} \mathrm {sgn}\left (b x + a\right ) + 360 \, a^{5} b x e^{5} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 428, normalized size = 1.44 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (10 b^{6} e^{6} x^{6}+72 a \,b^{5} e^{6} x^{5}-12 b^{6} d \,e^{5} x^{5}+225 a^{2} b^{4} e^{6} x^{4}-90 a \,b^{5} d \,e^{5} x^{4}+15 b^{6} d^{2} e^{4} x^{4}+400 a^{3} b^{3} e^{6} x^{3}-300 a^{2} b^{4} d \,e^{5} x^{3}+120 a \,b^{5} d^{2} e^{4} x^{3}-20 b^{6} d^{3} e^{3} x^{3}+450 a^{4} b^{2} e^{6} x^{2}-600 a^{3} b^{3} d \,e^{5} x^{2}+450 a^{2} b^{4} d^{2} e^{4} x^{2}-180 a \,b^{5} d^{3} e^{3} x^{2}+30 b^{6} d^{4} e^{2} x^{2}+60 a^{6} e^{6} \ln \left (e x +d \right )-360 a^{5} b d \,e^{5} \ln \left (e x +d \right )+360 a^{5} b \,e^{6} x +900 a^{4} b^{2} d^{2} e^{4} \ln \left (e x +d \right )-900 a^{4} b^{2} d \,e^{5} x -1200 a^{3} b^{3} d^{3} e^{3} \ln \left (e x +d \right )+1200 a^{3} b^{3} d^{2} e^{4} x +900 a^{2} b^{4} d^{4} e^{2} \ln \left (e x +d \right )-900 a^{2} b^{4} d^{3} e^{3} x -360 a \,b^{5} d^{5} e \ln \left (e x +d \right )+360 a \,b^{5} d^{4} e^{2} x +60 b^{6} d^{6} \ln \left (e x +d \right )-60 b^{6} d^{5} e x \right )}{60 \left (b x +a \right )^{5} e^{7}} \end {gather*}
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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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 )}^{5/2}}{d+e\,x} \,d x \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 {5}{2}}}{d + e x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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