Optimal. Leaf size=172 \[ \frac {3 e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)}{7 b^4}+\frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^2}{2 b^4}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)^3}{5 b^4}+\frac {e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^4} \]
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Rubi [A] time = 0.18, antiderivative size = 172, 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 {3 e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)}{7 b^4}+\frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^2}{2 b^4}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)^3}{5 b^4}+\frac {e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^4} \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)^3 \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)^3 \, 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)^3 \, 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)^3 (a+b x)^4}{b^3}+\frac {3 e (b d-a e)^2 (a+b x)^5}{b^3}+\frac {3 e^2 (b d-a e) (a+b x)^6}{b^3}+\frac {e^3 (a+b x)^7}{b^3}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e)^3 (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 b^4}+\frac {e (b d-a e)^2 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{2 b^4}+\frac {3 e^2 (b d-a e) (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^4}+\frac {e^3 (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^4}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 212, normalized size = 1.23 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (70 a^4 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+56 a^3 b x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+28 a^2 b^2 x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+8 a b^3 x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )+b^4 x^4 \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )\right )}{280 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 1.95, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (d+e x)^3 \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 = 225, normalized size = 1.31 \begin {gather*} \frac {1}{8} \, b^{4} e^{3} x^{8} + a^{4} d^{3} x + \frac {1}{7} \, {\left (3 \, b^{4} d e^{2} + 4 \, a b^{3} e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (b^{4} d^{2} e + 4 \, a b^{3} d e^{2} + 2 \, a^{2} b^{2} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{3} + 12 \, a b^{3} d^{2} e + 18 \, a^{2} b^{2} d e^{2} + 4 \, a^{3} b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, a b^{3} d^{3} + 18 \, a^{2} b^{2} d^{2} e + 12 \, a^{3} b d e^{2} + a^{4} e^{3}\right )} x^{4} + {\left (2 \, a^{2} b^{2} d^{3} + 4 \, a^{3} b d^{2} e + a^{4} d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d^{3} + 3 \, a^{4} d^{2} e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 360, normalized size = 2.09 \begin {gather*} \frac {1}{8} \, b^{4} x^{8} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{7} \, b^{4} d x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, b^{4} d^{2} x^{6} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, b^{4} d^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{7} \, a b^{3} x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, a b^{3} d x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {12}{5} \, a b^{3} d^{2} x^{5} e \mathrm {sgn}\left (b x + a\right ) + a b^{3} d^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + a^{2} b^{2} x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {18}{5} \, a^{2} b^{2} d x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{2} \, a^{2} b^{2} d^{2} x^{4} e \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{2} b^{2} d^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{5} \, a^{3} b x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{3} b d x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, a^{3} b d^{2} x^{3} e \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{3} b d^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, a^{4} x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} d x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a^{4} d^{2} x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{4} d^{3} 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 = 264, normalized size = 1.53 \begin {gather*} \frac {\left (35 e^{3} b^{4} x^{7}+160 x^{6} e^{3} a \,b^{3}+120 x^{6} d \,e^{2} b^{4}+280 x^{5} e^{3} a^{2} b^{2}+560 x^{5} d \,e^{2} a \,b^{3}+140 x^{5} d^{2} e \,b^{4}+224 x^{4} e^{3} a^{3} b +1008 x^{4} d \,e^{2} a^{2} b^{2}+672 x^{4} d^{2} e a \,b^{3}+56 x^{4} d^{3} b^{4}+70 x^{3} e^{3} a^{4}+840 x^{3} d \,e^{2} a^{3} b +1260 x^{3} d^{2} e \,a^{2} b^{2}+280 x^{3} d^{3} a \,b^{3}+280 a^{4} d \,e^{2} x^{2}+1120 a^{3} b \,d^{2} e \,x^{2}+560 a^{2} b^{2} d^{3} x^{2}+420 x \,d^{2} e \,a^{4}+560 x \,d^{3} a^{3} b +280 d^{3} a^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} x}{280 \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 = 0.65, size = 693, normalized size = 4.03 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} e^{3} x^{3}}{8 \, b} + \frac {1}{4} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a d^{3} x + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{4} e^{3} x}{4 \, b^{3}} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a e^{3} x^{2}}{56 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} d^{3}}{4 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{5} e^{3}}{4 \, b^{4}} + \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} e^{3} x}{56 \, b^{3}} - \frac {69 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} e^{3}}{280 \, b^{4}} - \frac {{\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3} x}{4 \, b^{3}} + \frac {3 \, {\left (b d^{2} e + a d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} x}{4 \, b^{2}} - \frac {{\left (b d^{3} + 3 \, a d^{2} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a x}{4 \, b} + \frac {{\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} x^{2}}{7 \, b^{2}} - \frac {{\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{4}}{4 \, b^{4}} + \frac {3 \, {\left (b d^{2} e + a d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3}}{4 \, b^{3}} - \frac {{\left (b d^{3} + 3 \, a d^{2} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2}}{4 \, b^{2}} - \frac {3 \, {\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a x}{14 \, b^{3}} + \frac {{\left (b d^{2} e + a d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} x}{2 \, b^{2}} + \frac {17 \, {\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2}}{70 \, b^{4}} - \frac {7 \, {\left (b d^{2} e + a d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a}{10 \, b^{3}} + \frac {{\left (b d^{3} + 3 \, a d^{2} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}}{5 \, b^{2}} \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 = -0.01 \begin {gather*} \int \left (a+b\,x\right )\,{\left (d+e\,x\right )}^3\,{\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 )^{3} \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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