Optimal. Leaf size=254 \[ \frac {3 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^2}{4 e^5 (a+b x)}-\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)^3}{7 e^5 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^4}{6 e^5 (a+b x)}+\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{10}}{10 e^5 (a+b x)}-\frac {4 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)}{9 e^5 (a+b x)} \]
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Rubi [A] time = 0.27, antiderivative size = 254, 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^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{10}}{10 e^5 (a+b x)}-\frac {4 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)}{9 e^5 (a+b x)}+\frac {3 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^2}{4 e^5 (a+b x)}-\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)^3}{7 e^5 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^4}{6 e^5 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 43
Rule 770
Rubi steps
\begin {align*} \int (a+b x) (d+e x)^5 \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 (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 (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 (d+e x)^5}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^6}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^7}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^8}{e^4}+\frac {b^4 (d+e x)^9}{e^4}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e)^4 (d+e x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^5 (a+b x)}-\frac {4 b (b d-a e)^3 (d+e x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}+\frac {3 b^2 (b d-a e)^2 (d+e x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x)}-\frac {4 b^3 (b d-a e) (d+e x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}+\frac {b^4 (d+e x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{10 e^5 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 322, normalized size = 1.27 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (210 a^4 \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+120 a^3 b x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )+45 a^2 b^2 x^2 \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )+10 a b^3 x^3 \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )+b^4 x^4 \left (252 d^5+1050 d^4 e x+1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+700 d e^4 x^4+126 e^5 x^5\right )\right )}{1260 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 3.23, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 360, normalized size = 1.42 \begin {gather*} \frac {1}{10} \, b^{4} e^{5} x^{10} + a^{4} d^{5} x + \frac {1}{9} \, {\left (5 \, b^{4} d e^{4} + 4 \, a b^{3} e^{5}\right )} x^{9} + \frac {1}{4} \, {\left (5 \, b^{4} d^{2} e^{3} + 10 \, a b^{3} d e^{4} + 3 \, a^{2} b^{2} e^{5}\right )} x^{8} + \frac {2}{7} \, {\left (5 \, b^{4} d^{3} e^{2} + 20 \, a b^{3} d^{2} e^{3} + 15 \, a^{2} b^{2} d e^{4} + 2 \, a^{3} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (5 \, b^{4} d^{4} e + 40 \, a b^{3} d^{3} e^{2} + 60 \, a^{2} b^{2} d^{2} e^{3} + 20 \, a^{3} b d e^{4} + a^{4} e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{5} + 20 \, a b^{3} d^{4} e + 60 \, a^{2} b^{2} d^{3} e^{2} + 40 \, a^{3} b d^{2} e^{3} + 5 \, a^{4} d e^{4}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, a b^{3} d^{5} + 15 \, a^{2} b^{2} d^{4} e + 20 \, a^{3} b d^{3} e^{2} + 5 \, a^{4} d^{2} e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (3 \, a^{2} b^{2} d^{5} + 10 \, a^{3} b d^{4} e + 5 \, a^{4} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d^{5} + 5 \, a^{4} d^{4} e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 561, normalized size = 2.21 \begin {gather*} \frac {1}{10} \, b^{4} x^{10} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{9} \, b^{4} d x^{9} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, b^{4} d^{2} x^{8} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{7} \, b^{4} d^{3} x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{6} \, b^{4} d^{4} x^{6} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, b^{4} d^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{9} \, a b^{3} x^{9} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a b^{3} d x^{8} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {40}{7} \, a b^{3} d^{2} x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{3} \, a b^{3} d^{3} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, a b^{3} d^{4} x^{5} e \mathrm {sgn}\left (b x + a\right ) + a b^{3} d^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{4} \, a^{2} b^{2} x^{8} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {30}{7} \, a^{2} b^{2} d x^{7} e^{4} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{2} d^{2} x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + 12 \, a^{2} b^{2} d^{3} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{2} \, a^{2} b^{2} d^{4} x^{4} e \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{2} b^{2} d^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{7} \, a^{3} b x^{7} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{3} b d x^{6} e^{4} \mathrm {sgn}\left (b x + a\right ) + 8 \, a^{3} b d^{2} x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{3} b d^{3} x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{3} \, a^{3} b d^{4} x^{3} e \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{3} b d^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, a^{4} x^{6} e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{4} d x^{5} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{4} d^{2} x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{4} d^{3} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{4} d^{4} x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{4} d^{5} x \mathrm {sgn}\left (b x + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 414, normalized size = 1.63 \begin {gather*} \frac {\left (126 b^{4} e^{5} x^{9}+560 x^{8} a \,b^{3} e^{5}+700 x^{8} b^{4} d \,e^{4}+945 x^{7} a^{2} b^{2} e^{5}+3150 x^{7} a \,b^{3} d \,e^{4}+1575 x^{7} b^{4} d^{2} e^{3}+720 x^{6} a^{3} b \,e^{5}+5400 x^{6} a^{2} b^{2} d \,e^{4}+7200 x^{6} a \,b^{3} d^{2} e^{3}+1800 x^{6} b^{4} d^{3} e^{2}+210 x^{5} a^{4} e^{5}+4200 x^{5} a^{3} b d \,e^{4}+12600 x^{5} a^{2} b^{2} d^{2} e^{3}+8400 x^{5} a \,b^{3} d^{3} e^{2}+1050 x^{5} b^{4} d^{4} e +1260 x^{4} a^{4} d \,e^{4}+10080 x^{4} a^{3} b \,d^{2} e^{3}+15120 x^{4} a^{2} b^{2} d^{3} e^{2}+5040 x^{4} a \,b^{3} d^{4} e +252 x^{4} b^{4} d^{5}+3150 x^{3} a^{4} d^{2} e^{3}+12600 x^{3} a^{3} b \,d^{3} e^{2}+9450 x^{3} a^{2} b^{2} d^{4} e +1260 x^{3} a \,b^{3} d^{5}+4200 x^{2} a^{4} d^{3} e^{2}+8400 x^{2} a^{3} b \,d^{4} e +2520 x^{2} a^{2} b^{2} d^{5}+3150 x \,a^{4} d^{4} e +2520 x \,a^{3} b \,d^{5}+1260 a^{4} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} x}{1260 \left (b x +a \right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.77, size = 1323, normalized size = 5.21
result too large to display
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 (a+b\,x\right )\,{\left (d+e\,x\right )}^5\,{\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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b x\right ) \left (d + e x\right )^{5} \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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