Optimal. Leaf size=298 \[ -\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^8 (-3 a B e-A b e+4 b B d)}{8 e^5 (a+b x)}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{6 e^5 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e)^3 (B d-A e)}{5 e^5 (a+b x)}+\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^9}{9 e^5 (a+b x)} \]
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Rubi [A] time = 0.40, antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {770, 77} \begin {gather*} -\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^8 (-3 a B e-A b e+4 b B d)}{8 e^5 (a+b x)}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e) (-a B e-A b e+2 b B d)}{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)^2 (-a B e-3 A b e+4 b B d)}{6 e^5 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e)^3 (B d-A e)}{5 e^5 (a+b x)}+\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^9}{9 e^5 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 770
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 \left (a b+b^2 x\right )^3 (A+B x) (d+e x)^4 \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^3 (b d-a e)^3 (-B d+A e) (d+e x)^4}{e^4}+\frac {b^3 (b d-a e)^2 (-4 b B d+3 A b e+a B e) (d+e x)^5}{e^4}-\frac {3 b^4 (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^6}{e^4}+\frac {b^5 (-4 b B d+A b e+3 a B e) (d+e x)^7}{e^4}+\frac {b^6 B (d+e x)^8}{e^4}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {(b d-a e)^3 (B d-A e) (d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x)}-\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^5 (a+b x)}+\frac {3 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}-\frac {b^2 (4 b B d-A b e-3 a B e) (d+e x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^5 (a+b x)}+\frac {b^3 B (d+e x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 410, normalized size = 1.38 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (84 a^3 \left (6 A \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 )+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 )\right )+36 a^2 b x \left (7 A \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 )+2 B x \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 )\right )+9 a b^2 x^2 \left (8 A \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 )+3 B x \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 )\right )+b^3 x^3 \left (9 A \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 )+4 B x \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 )\right )}{2520 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 5.33, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x)^4 \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 = 0.42, size = 425, normalized size = 1.43 \begin {gather*} \frac {1}{9} \, B b^{3} e^{4} x^{9} + A a^{3} d^{4} x + \frac {1}{8} \, {\left (4 \, B b^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (6 \, B b^{3} d^{2} e^{2} + 4 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (4 \, B b^{3} d^{3} e + 6 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} d^{4} + A a^{3} e^{4} + 4 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 18 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 4 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, A a^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} e + 6 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a^{3} d^{2} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{4} + 4 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{3} d^{3} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4}\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 758, normalized size = 2.54 \begin {gather*} \frac {1}{9} \, B b^{3} x^{9} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, B b^{3} d x^{8} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{7} \, B b^{3} d^{2} x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, B b^{3} d^{3} x^{6} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, B b^{3} d^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{8} \, B a b^{2} x^{8} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{8} \, A b^{3} x^{8} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {12}{7} \, B a b^{2} d x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{7} \, A b^{3} d x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, B a b^{2} d^{2} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + A b^{3} d^{2} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {12}{5} \, B a b^{2} d^{3} x^{5} e \mathrm {sgn}\left (b x + a\right ) + \frac {4}{5} \, A b^{3} d^{3} x^{5} e \mathrm {sgn}\left (b x + a\right ) + \frac {3}{4} \, B a b^{2} d^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, A b^{3} d^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{7} \, B a^{2} b x^{7} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{7} \, A a b^{2} x^{7} e^{4} \mathrm {sgn}\left (b x + a\right ) + 2 \, B a^{2} b d x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a b^{2} d x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {18}{5} \, B a^{2} b d^{2} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {18}{5} \, A a b^{2} d^{2} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, B a^{2} b d^{3} x^{4} e \mathrm {sgn}\left (b x + a\right ) + 3 \, A a b^{2} d^{3} x^{4} e \mathrm {sgn}\left (b x + a\right ) + B a^{2} b d^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + A a b^{2} d^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, B a^{3} x^{6} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, A a^{2} b x^{6} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{5} \, B a^{3} d x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {12}{5} \, A a^{2} b d x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, B a^{3} d^{2} x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{2} \, A a^{2} b d^{2} x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{3} \, B a^{3} d^{3} x^{3} e \mathrm {sgn}\left (b x + a\right ) + 4 \, A a^{2} b d^{3} x^{3} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, B a^{3} d^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, A a^{2} b d^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, A a^{3} x^{5} e^{4} \mathrm {sgn}\left (b x + a\right ) + A a^{3} d x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a^{3} d^{2} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a^{3} d^{3} x^{2} e \mathrm {sgn}\left (b x + a\right ) + A a^{3} 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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maple [B] time = 0.05, size = 552, normalized size = 1.85 \begin {gather*} \frac {\left (280 b^{3} B \,e^{4} x^{8}+315 x^{7} A \,b^{3} e^{4}+945 x^{7} B a \,b^{2} e^{4}+1260 x^{7} b^{3} B d \,e^{3}+1080 x^{6} A a \,b^{2} e^{4}+1440 x^{6} A \,b^{3} d \,e^{3}+1080 x^{6} B \,a^{2} b \,e^{4}+4320 x^{6} B a \,b^{2} d \,e^{3}+2160 x^{6} b^{3} B \,d^{2} e^{2}+1260 x^{5} A \,a^{2} b \,e^{4}+5040 x^{5} A a \,b^{2} d \,e^{3}+2520 x^{5} A \,b^{3} d^{2} e^{2}+420 x^{5} B \,a^{3} e^{4}+5040 x^{5} B \,a^{2} b d \,e^{3}+7560 x^{5} B a \,b^{2} d^{2} e^{2}+1680 x^{5} b^{3} B \,d^{3} e +504 x^{4} A \,a^{3} e^{4}+6048 x^{4} A \,a^{2} b d \,e^{3}+9072 x^{4} A a \,b^{2} d^{2} e^{2}+2016 x^{4} A \,b^{3} d^{3} e +2016 x^{4} B \,a^{3} d \,e^{3}+9072 x^{4} B \,a^{2} b \,d^{2} e^{2}+6048 x^{4} B a \,b^{2} d^{3} e +504 x^{4} b^{3} B \,d^{4}+2520 x^{3} A \,a^{3} d \,e^{3}+11340 x^{3} A \,a^{2} b \,d^{2} e^{2}+7560 x^{3} A a \,b^{2} d^{3} e +630 x^{3} A \,b^{3} d^{4}+3780 x^{3} B \,a^{3} d^{2} e^{2}+7560 x^{3} B \,a^{2} b \,d^{3} e +1890 x^{3} B a \,b^{2} d^{4}+5040 x^{2} A \,a^{3} d^{2} e^{2}+10080 x^{2} A \,a^{2} b \,d^{3} e +2520 x^{2} A a \,b^{2} d^{4}+3360 x^{2} B \,a^{3} d^{3} e +2520 x^{2} B \,a^{2} b \,d^{4}+5040 x A \,a^{3} d^{3} e +3780 x A \,a^{2} b \,d^{4}+1260 x B \,a^{3} d^{4}+2520 A \,a^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} x}{2520 \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 = 0.79, size = 1004, normalized size = 3.37
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 )}^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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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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