Optimal. Leaf size=219 \[ \frac {(b d-a e)^4 (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 b^5}+\frac {2 e (b d-a e)^3 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{3 b^5}+\frac {6 e^2 (b d-a e)^2 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^5}+\frac {e^3 (b d-a e) (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{2 b^5}+\frac {e^4 (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^5} \]
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Rubi [A]
time = 0.15, antiderivative size = 219, 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 {e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)}{2 b^5}+\frac {6 e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^2}{7 b^5}+\frac {2 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^3}{3 b^5}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)^4}{5 b^5}+\frac {e^4 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 45
Rule 784
Rubi steps
\begin {align*} \int (a+b x) (d+e x)^4 \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)^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 (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 d-a e)^4 (a+b x)^4}{b^4}+\frac {4 e (b d-a e)^3 (a+b x)^5}{b^4}+\frac {6 e^2 (b d-a e)^2 (a+b x)^6}{b^4}+\frac {4 e^3 (b d-a e) (a+b x)^7}{b^4}+\frac {e^4 (a+b x)^8}{b^4}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e)^4 (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 b^5}+\frac {2 e (b d-a e)^3 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{3 b^5}+\frac {6 e^2 (b d-a e)^2 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^5}+\frac {e^3 (b d-a e) (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{2 b^5}+\frac {e^4 (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^5}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 267, normalized size = 1.22 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (126 a^4 \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+84 a^3 b x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )+36 a^2 b^2 x^2 \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )+9 a b^3 x^3 \left (70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right )+b^4 x^4 \left (126 d^4+420 d^3 e x+540 d^2 e^2 x^2+315 d e^3 x^3+70 e^4 x^4\right )\right )}{630 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(338\) vs.
\(2(154)=308\).
time = 0.06, size = 339, normalized size = 1.55
method | result | size |
gosper | \(\frac {x \left (70 b^{4} e^{4} x^{8}+315 x^{7} e^{4} a \,b^{3}+315 x^{7} d \,e^{3} b^{4}+540 x^{6} e^{4} a^{2} b^{2}+1440 x^{6} d \,e^{3} a \,b^{3}+540 x^{6} d^{2} e^{2} b^{4}+420 x^{5} e^{4} a^{3} b +2520 x^{5} d \,e^{3} a^{2} b^{2}+2520 x^{5} d^{2} e^{2} a \,b^{3}+420 x^{5} d^{3} e \,b^{4}+126 x^{4} a^{4} e^{4}+2016 x^{4} a^{3} b d \,e^{3}+4536 x^{4} a^{2} b^{2} d^{2} e^{2}+2016 x^{4} a \,b^{3} d^{3} e +126 x^{4} b^{4} d^{4}+630 a^{4} d \,e^{3} x^{3}+3780 a^{3} b \,d^{2} e^{2} x^{3}+3780 a^{2} b^{2} d^{3} e \,x^{3}+630 a \,b^{3} d^{4} x^{3}+1260 x^{2} d^{2} e^{2} a^{4}+3360 x^{2} d^{3} e \,a^{3} b +1260 x^{2} d^{4} a^{2} b^{2}+1260 a^{4} d^{3} e x +1260 a^{3} b \,d^{4} x +630 d^{4} a^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{630 \left (b x +a \right )^{3}}\) | \(339\) |
default | \(\frac {x \left (70 b^{4} e^{4} x^{8}+315 x^{7} e^{4} a \,b^{3}+315 x^{7} d \,e^{3} b^{4}+540 x^{6} e^{4} a^{2} b^{2}+1440 x^{6} d \,e^{3} a \,b^{3}+540 x^{6} d^{2} e^{2} b^{4}+420 x^{5} e^{4} a^{3} b +2520 x^{5} d \,e^{3} a^{2} b^{2}+2520 x^{5} d^{2} e^{2} a \,b^{3}+420 x^{5} d^{3} e \,b^{4}+126 x^{4} a^{4} e^{4}+2016 x^{4} a^{3} b d \,e^{3}+4536 x^{4} a^{2} b^{2} d^{2} e^{2}+2016 x^{4} a \,b^{3} d^{3} e +126 x^{4} b^{4} d^{4}+630 a^{4} d \,e^{3} x^{3}+3780 a^{3} b \,d^{2} e^{2} x^{3}+3780 a^{2} b^{2} d^{3} e \,x^{3}+630 a \,b^{3} d^{4} x^{3}+1260 x^{2} d^{2} e^{2} a^{4}+3360 x^{2} d^{3} e \,a^{3} b +1260 x^{2} d^{4} a^{2} b^{2}+1260 a^{4} d^{3} e x +1260 a^{3} b \,d^{4} x +630 d^{4} a^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{630 \left (b x +a \right )^{3}}\) | \(339\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{4} e^{4} x^{9}}{9 b x +9 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 e^{4} a \,b^{3}+4 d \,e^{3} b^{4}\right ) x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (6 e^{4} a^{2} b^{2}+16 d \,e^{3} a \,b^{3}+6 d^{2} e^{2} b^{4}\right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 e^{4} a^{3} b +24 d \,e^{3} a^{2} b^{2}+24 d^{2} e^{2} a \,b^{3}+4 d^{3} e \,b^{4}\right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a^{4} e^{4}+16 a^{3} b d \,e^{3}+36 a^{2} b^{2} d^{2} e^{2}+16 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 d \,e^{3} a^{4}+24 d^{2} e^{2} a^{3} b +24 d^{3} e \,a^{2} b^{2}+4 d^{4} a \,b^{3}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (6 d^{2} e^{2} a^{4}+16 d^{3} e \,a^{3} b +6 d^{4} a^{2} b^{2}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (4 d^{3} e \,a^{4}+4 d^{4} a^{3} b \right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, d^{4} a^{4} x}{b x +a}\) | \(439\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 972 vs.
\(2 (156) = 312\).
time = 0.30, size = 972, normalized size = 4.44 \begin {gather*} \frac {1}{4} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a d^{4} x + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} d^{4}}{4 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} x^{4} e^{4}}{9 \, b} - \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a x^{3} e^{4}}{72 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (4 \, b d e^{3} + a e^{4}\right )} x^{3}}{8 \, b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{5} x e^{4}}{4 \, b^{4}} + \frac {37 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} x^{2} e^{4}}{168 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (4 \, b d e^{3} + a e^{4}\right )} a^{4} x}{4 \, b^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} a^{3} x}{2 \, b^{3}} + \frac {{\left (2 \, b d^{3} e + 3 \, a d^{2} e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} x}{2 \, b^{2}} - \frac {{\left (b d^{4} + 4 \, a d^{3} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a x}{4 \, b} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (4 \, b d e^{3} + a e^{4}\right )} a x^{2}}{56 \, b^{3}} + \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} x^{2}}{7 \, b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{6} e^{4}}{4 \, b^{5}} - \frac {121 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} x e^{4}}{504 \, b^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (4 \, b d e^{3} + a e^{4}\right )} a^{5}}{4 \, b^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} a^{4}}{2 \, b^{4}} + \frac {{\left (2 \, b d^{3} e + 3 \, a d^{2} e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3}}{2 \, b^{3}} - \frac {{\left (b d^{4} + 4 \, a d^{3} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2}}{4 \, b^{2}} + \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (4 \, b d e^{3} + a e^{4}\right )} a^{2} x}{56 \, b^{4}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} a x}{7 \, b^{3}} + \frac {{\left (2 \, b d^{3} e + 3 \, a d^{2} e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} x}{3 \, b^{2}} + \frac {125 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4} e^{4}}{504 \, b^{5}} - \frac {69 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (4 \, b d e^{3} + a e^{4}\right )} a^{3}}{280 \, b^{5}} + \frac {17 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} a^{2}}{35 \, b^{4}} - \frac {7 \, {\left (2 \, b d^{3} e + 3 \, a d^{2} e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a}{15 \, b^{3}} + \frac {{\left (b d^{4} + 4 \, a d^{3} 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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Fricas [A]
time = 2.54, size = 292, normalized size = 1.33 \begin {gather*} \frac {1}{5} \, b^{4} d^{4} x^{5} + a b^{3} d^{4} x^{4} + 2 \, a^{2} b^{2} d^{4} x^{3} + 2 \, a^{3} b d^{4} x^{2} + a^{4} d^{4} x + \frac {1}{630} \, {\left (70 \, b^{4} x^{9} + 315 \, a b^{3} x^{8} + 540 \, a^{2} b^{2} x^{7} + 420 \, a^{3} b x^{6} + 126 \, a^{4} x^{5}\right )} e^{4} + \frac {1}{70} \, {\left (35 \, b^{4} d x^{8} + 160 \, a b^{3} d x^{7} + 280 \, a^{2} b^{2} d x^{6} + 224 \, a^{3} b d x^{5} + 70 \, a^{4} d x^{4}\right )} e^{3} + \frac {2}{35} \, {\left (15 \, b^{4} d^{2} x^{7} + 70 \, a b^{3} d^{2} x^{6} + 126 \, a^{2} b^{2} d^{2} x^{5} + 105 \, a^{3} b d^{2} x^{4} + 35 \, a^{4} d^{2} x^{3}\right )} e^{2} + \frac {2}{15} \, {\left (5 \, b^{4} d^{3} x^{6} + 24 \, a b^{3} d^{3} x^{5} + 45 \, a^{2} b^{2} d^{3} x^{4} + 40 \, a^{3} b d^{3} x^{3} + 15 \, a^{4} d^{3} x^{2}\right )} e \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 )^{4} \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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 461 vs.
\(2 (156) = 312\).
time = 0.80, size = 461, normalized size = 2.11 \begin {gather*} \frac {1}{9} \, b^{4} x^{9} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, b^{4} d x^{8} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{7} \, b^{4} d^{2} x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, b^{4} d^{3} x^{6} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, b^{4} d^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, a b^{3} x^{8} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {16}{7} \, a b^{3} d x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + 4 \, a b^{3} d^{2} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {16}{5} \, a b^{3} d^{3} x^{5} e \mathrm {sgn}\left (b x + a\right ) + a b^{3} d^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{7} \, a^{2} b^{2} x^{7} e^{4} \mathrm {sgn}\left (b x + a\right ) + 4 \, a^{2} b^{2} d x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {36}{5} \, a^{2} b^{2} d^{2} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{3} x^{4} e \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{2} b^{2} d^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, a^{3} b x^{6} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {16}{5} \, a^{3} b d x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{3} b d^{2} x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {16}{3} \, a^{3} b d^{3} x^{3} e \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{3} b d^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, a^{4} x^{5} e^{4} \mathrm {sgn}\left (b x + a\right ) + a^{4} d x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{4} d^{2} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{4} d^{3} x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{4} d^{4} x \mathrm {sgn}\left (b x + a\right ) \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 \left (a+b\,x\right )\,{\left (d+e\,x\right )}^4\,{\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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