Optimal. Leaf size=298 \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^9 (-3 a B e-A b e+4 b B d)}{9 e^5 (a+b x)}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e) (-a B e-A b e+2 b B d)}{8 e^5 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^3 (B d-A e)}{6 e^5 (a+b x)}+\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{10}}{10 e^5 (a+b x)} \]
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Rubi [A] time = 0.508242, antiderivative size = 298, normalized size of antiderivative = 1., 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} \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^9 (-3 a B e-A b e+4 b B d)}{9 e^5 (a+b x)}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e) (-a B e-A b e+2 b B d)}{8 e^5 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^3 (B d-A e)}{6 e^5 (a+b x)}+\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{10}}{10 e^5 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 77
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 \left (a b+b^2 x\right )^3 (A+B x) (d+e x)^5 \, 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)^5}{e^4}+\frac{b^3 (b d-a e)^2 (-4 b B d+3 A b e+a B e) (d+e x)^6}{e^4}-\frac{3 b^4 (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^7}{e^4}+\frac{b^5 (-4 b B d+A b e+3 a B e) (d+e x)^8}{e^4}+\frac{b^6 B (d+e x)^9}{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)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 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)^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 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)^8 \sqrt{a^2+2 a b x+b^2 x^2}}{8 e^5 (a+b x)}-\frac{b^2 (4 b B d-A b e-3 a B e) (d+e x)^9 \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}+\frac{b^3 B (d+e x)^{10} \sqrt{a^2+2 a b x+b^2 x^2}}{10 e^5 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.223912, size = 496, normalized size = 1.66 \[ \frac{x \sqrt{(a+b x)^2} \left (45 a^2 b x \left (4 A \left (105 d^3 e^2 x^2+84 d^2 e^3 x^3+70 d^4 e x+21 d^5+35 d e^4 x^4+6 e^5 x^5\right )+B x \left (336 d^3 e^2 x^2+280 d^2 e^3 x^3+210 d^4 e x+56 d^5+120 d e^4 x^4+21 e^5 x^5\right )\right )+60 a^3 \left (7 A \left (20 d^3 e^2 x^2+15 d^2 e^3 x^3+15 d^4 e x+6 d^5+6 d e^4 x^4+e^5 x^5\right )+B x \left (105 d^3 e^2 x^2+84 d^2 e^3 x^3+70 d^4 e x+21 d^5+35 d e^4 x^4+6 e^5 x^5\right )\right )+15 a b^2 x^2 \left (3 A \left (336 d^3 e^2 x^2+280 d^2 e^3 x^3+210 d^4 e x+56 d^5+120 d e^4 x^4+21 e^5 x^5\right )+B x \left (840 d^3 e^2 x^2+720 d^2 e^3 x^3+504 d^4 e x+126 d^5+315 d e^4 x^4+56 e^5 x^5\right )\right )+b^3 x^3 \left (5 A \left (840 d^3 e^2 x^2+720 d^2 e^3 x^3+504 d^4 e x+126 d^5+315 d e^4 x^4+56 e^5 x^5\right )+2 B x \left (1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+1050 d^4 e x+252 d^5+700 d e^4 x^4+126 e^5 x^5\right )\right )\right )}{2520 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 676, normalized size = 2.3 \begin{align*}{\frac{x \left ( 252\,{b}^{3}B{e}^{5}{x}^{9}+280\,{x}^{8}A{b}^{3}{e}^{5}+840\,{x}^{8}Ba{b}^{2}{e}^{5}+1400\,{x}^{8}{b}^{3}Bd{e}^{4}+945\,{x}^{7}Aa{b}^{2}{e}^{5}+1575\,{x}^{7}A{b}^{3}d{e}^{4}+945\,{x}^{7}B{a}^{2}b{e}^{5}+4725\,{x}^{7}Ba{b}^{2}d{e}^{4}+3150\,{x}^{7}{b}^{3}B{d}^{2}{e}^{3}+1080\,{x}^{6}A{a}^{2}b{e}^{5}+5400\,{x}^{6}Aa{b}^{2}d{e}^{4}+3600\,{x}^{6}A{b}^{3}{d}^{2}{e}^{3}+360\,{x}^{6}B{e}^{5}{a}^{3}+5400\,{x}^{6}B{a}^{2}bd{e}^{4}+10800\,{x}^{6}Ba{b}^{2}{d}^{2}{e}^{3}+3600\,{x}^{6}{b}^{3}B{d}^{3}{e}^{2}+420\,{x}^{5}A{a}^{3}{e}^{5}+6300\,{x}^{5}A{a}^{2}bd{e}^{4}+12600\,{x}^{5}Aa{b}^{2}{d}^{2}{e}^{3}+4200\,{x}^{5}A{b}^{3}{d}^{3}{e}^{2}+2100\,{x}^{5}B{a}^{3}d{e}^{4}+12600\,{x}^{5}B{a}^{2}b{d}^{2}{e}^{3}+12600\,{x}^{5}Ba{b}^{2}{d}^{3}{e}^{2}+2100\,{x}^{5}{b}^{3}B{d}^{4}e+2520\,{x}^{4}A{a}^{3}d{e}^{4}+15120\,{x}^{4}A{a}^{2}b{d}^{2}{e}^{3}+15120\,{x}^{4}Aa{b}^{2}{d}^{3}{e}^{2}+2520\,{x}^{4}A{b}^{3}{d}^{4}e+5040\,{x}^{4}B{a}^{3}{d}^{2}{e}^{3}+15120\,{x}^{4}B{a}^{2}b{d}^{3}{e}^{2}+7560\,{x}^{4}Ba{b}^{2}{d}^{4}e+504\,{x}^{4}{b}^{3}B{d}^{5}+6300\,{x}^{3}A{a}^{3}{d}^{2}{e}^{3}+18900\,{x}^{3}A{a}^{2}b{d}^{3}{e}^{2}+9450\,{x}^{3}Aa{b}^{2}{d}^{4}e+630\,{x}^{3}A{b}^{3}{d}^{5}+6300\,{x}^{3}B{a}^{3}{d}^{3}{e}^{2}+9450\,{x}^{3}B{a}^{2}b{d}^{4}e+1890\,{x}^{3}Ba{b}^{2}{d}^{5}+8400\,{x}^{2}A{a}^{3}{d}^{3}{e}^{2}+12600\,{x}^{2}A{a}^{2}b{d}^{4}e+2520\,{x}^{2}Aa{b}^{2}{d}^{5}+4200\,{x}^{2}B{a}^{3}{d}^{4}e+2520\,{x}^{2}B{a}^{2}b{d}^{5}+6300\,xA{a}^{3}{d}^{4}e+3780\,xA{a}^{2}b{d}^{5}+1260\,xB{a}^{3}{d}^{5}+2520\,A{a}^{3}{d}^{5} \right ) }{2520\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.61404, size = 1106, normalized size = 3.71 \begin{align*} \frac{1}{10} \, B b^{3} e^{5} x^{10} + A a^{3} d^{5} x + \frac{1}{9} \,{\left (5 \, B b^{3} d e^{4} +{\left (3 \, B a b^{2} + A b^{3}\right )} e^{5}\right )} x^{9} + \frac{1}{8} \,{\left (10 \, B b^{3} d^{2} e^{3} + 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{4} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} e^{5}\right )} x^{8} + \frac{1}{7} \,{\left (10 \, B b^{3} d^{3} e^{2} + 10 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{3} + 15 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{4} +{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (5 \, B b^{3} d^{4} e + A a^{3} e^{5} + 10 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{2} + 30 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{3} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (B b^{3} d^{5} + 5 \, A a^{3} d e^{4} + 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e + 30 \,{\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{2} + 10 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (10 \, A a^{3} d^{2} e^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} + 15 \,{\left (B a^{2} b + A a b^{2}\right )} d^{4} e + 10 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (10 \, A a^{3} d^{3} e^{2} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d^{5} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4} e\right )} x^{3} + \frac{1}{2} \,{\left (5 \, A a^{3} d^{4} e +{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17523, size = 1245, normalized size = 4.18 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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