Optimal. Leaf size=216 \[ -\frac{2 b^3 (d+e x)^{9/2} (-4 a B e-A b e+5 b B d)}{9 e^6}+\frac{4 b^2 (d+e x)^{7/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{7 e^6}-\frac{4 b (d+e x)^{5/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{5 e^6}+\frac{2 (d+e x)^{3/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{3 e^6}-\frac{2 \sqrt{d+e x} (b d-a e)^4 (B d-A e)}{e^6}+\frac{2 b^4 B (d+e x)^{11/2}}{11 e^6} \]
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Rubi [A] time = 0.0943628, antiderivative size = 216, 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 = {27, 77} \[ -\frac{2 b^3 (d+e x)^{9/2} (-4 a B e-A b e+5 b B d)}{9 e^6}+\frac{4 b^2 (d+e x)^{7/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{7 e^6}-\frac{4 b (d+e x)^{5/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{5 e^6}+\frac{2 (d+e x)^{3/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{3 e^6}-\frac{2 \sqrt{d+e x} (b d-a e)^4 (B d-A e)}{e^6}+\frac{2 b^4 B (d+e x)^{11/2}}{11 e^6} \]
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt{d+e x}} \, dx &=\int \frac{(a+b x)^4 (A+B x)}{\sqrt{d+e x}} \, dx\\ &=\int \left (\frac{(-b d+a e)^4 (-B d+A e)}{e^5 \sqrt{d+e x}}+\frac{(-b d+a e)^3 (-5 b B d+4 A b e+a B e) \sqrt{d+e x}}{e^5}+\frac{2 b (b d-a e)^2 (-5 b B d+3 A b e+2 a B e) (d+e x)^{3/2}}{e^5}-\frac{2 b^2 (b d-a e) (-5 b B d+2 A b e+3 a B e) (d+e x)^{5/2}}{e^5}+\frac{b^3 (-5 b B d+A b e+4 a B e) (d+e x)^{7/2}}{e^5}+\frac{b^4 B (d+e x)^{9/2}}{e^5}\right ) \, dx\\ &=-\frac{2 (b d-a e)^4 (B d-A e) \sqrt{d+e x}}{e^6}+\frac{2 (b d-a e)^3 (5 b B d-4 A b e-a B e) (d+e x)^{3/2}}{3 e^6}-\frac{4 b (b d-a e)^2 (5 b B d-3 A b e-2 a B e) (d+e x)^{5/2}}{5 e^6}+\frac{4 b^2 (b d-a e) (5 b B d-2 A b e-3 a B e) (d+e x)^{7/2}}{7 e^6}-\frac{2 b^3 (5 b B d-A b e-4 a B e) (d+e x)^{9/2}}{9 e^6}+\frac{2 b^4 B (d+e x)^{11/2}}{11 e^6}\\ \end{align*}
Mathematica [A] time = 0.128065, size = 183, normalized size = 0.85 \[ \frac{2 \sqrt{d+e x} \left (-385 b^3 (d+e x)^4 (-4 a B e-A b e+5 b B d)+990 b^2 (d+e x)^3 (b d-a e) (-3 a B e-2 A b e+5 b B d)-1386 b (d+e x)^2 (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)+1155 (d+e x) (b d-a e)^3 (-a B e-4 A b e+5 b B d)-3465 (b d-a e)^4 (B d-A e)+315 b^4 B (d+e x)^5\right )}{3465 e^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 469, normalized size = 2.2 \begin{align*}{\frac{630\,{b}^{4}B{x}^{5}{e}^{5}+770\,A{b}^{4}{e}^{5}{x}^{4}+3080\,Ba{b}^{3}{e}^{5}{x}^{4}-700\,B{b}^{4}d{e}^{4}{x}^{4}+3960\,Aa{b}^{3}{e}^{5}{x}^{3}-880\,A{b}^{4}d{e}^{4}{x}^{3}+5940\,B{a}^{2}{b}^{2}{e}^{5}{x}^{3}-3520\,Ba{b}^{3}d{e}^{4}{x}^{3}+800\,B{b}^{4}{d}^{2}{e}^{3}{x}^{3}+8316\,A{a}^{2}{b}^{2}{e}^{5}{x}^{2}-4752\,Aa{b}^{3}d{e}^{4}{x}^{2}+1056\,A{b}^{4}{d}^{2}{e}^{3}{x}^{2}+5544\,B{a}^{3}b{e}^{5}{x}^{2}-7128\,B{a}^{2}{b}^{2}d{e}^{4}{x}^{2}+4224\,Ba{b}^{3}{d}^{2}{e}^{3}{x}^{2}-960\,B{b}^{4}{d}^{3}{e}^{2}{x}^{2}+9240\,A{a}^{3}b{e}^{5}x-11088\,A{a}^{2}{b}^{2}d{e}^{4}x+6336\,Aa{b}^{3}{d}^{2}{e}^{3}x-1408\,A{b}^{4}{d}^{3}{e}^{2}x+2310\,B{a}^{4}{e}^{5}x-7392\,B{a}^{3}bd{e}^{4}x+9504\,B{a}^{2}{b}^{2}{d}^{2}{e}^{3}x-5632\,Ba{b}^{3}{d}^{3}{e}^{2}x+1280\,B{b}^{4}{d}^{4}ex+6930\,A{a}^{4}{e}^{5}-18480\,Ad{a}^{3}b{e}^{4}+22176\,A{d}^{2}{a}^{2}{b}^{2}{e}^{3}-12672\,Aa{b}^{3}{d}^{3}{e}^{2}+2816\,A{b}^{4}{d}^{4}e-4620\,B{a}^{4}d{e}^{4}+14784\,B{d}^{2}{a}^{3}b{e}^{3}-19008\,B{d}^{3}{a}^{2}{b}^{2}{e}^{2}+11264\,Ba{b}^{3}{d}^{4}e-2560\,{b}^{4}B{d}^{5}}{3465\,{e}^{6}}\sqrt{ex+d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.999845, size = 552, normalized size = 2.56 \begin{align*} \frac{2 \,{\left (315 \,{\left (e x + d\right )}^{\frac{11}{2}} B b^{4} - 385 \,{\left (5 \, B b^{4} d -{\left (4 \, B a b^{3} + A b^{4}\right )} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 990 \,{\left (5 \, B b^{4} d^{2} - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 1386 \,{\left (5 \, B b^{4} d^{3} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{2} -{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 1155 \,{\left (5 \, B b^{4} d^{4} - 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{2} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{4}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 3465 \,{\left (B b^{4} d^{5} - A a^{4} e^{5} -{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4}\right )} \sqrt{e x + d}\right )}}{3465 \, e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.42096, size = 915, normalized size = 4.24 \begin{align*} \frac{2 \,{\left (315 \, B b^{4} e^{5} x^{5} - 1280 \, B b^{4} d^{5} + 3465 \, A a^{4} e^{5} + 1408 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e - 3168 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 3696 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} - 2310 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} - 35 \,{\left (10 \, B b^{4} d e^{4} - 11 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 10 \,{\left (40 \, B b^{4} d^{2} e^{3} - 44 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 99 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} - 6 \,{\left (80 \, B b^{4} d^{3} e^{2} - 88 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 198 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} - 231 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} +{\left (640 \, B b^{4} d^{4} e - 704 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 1584 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 1848 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 1155 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 118.522, size = 1311, normalized size = 6.07 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15379, size = 679, normalized size = 3.14 \begin{align*} \frac{2}{3465} \,{\left (1155 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} B a^{4} e^{\left (-1\right )} + 4620 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} A a^{3} b e^{\left (-1\right )} + 924 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} B a^{3} b e^{\left (-2\right )} + 1386 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} A a^{2} b^{2} e^{\left (-2\right )} + 594 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{x e + d} d^{3}\right )} B a^{2} b^{2} e^{\left (-3\right )} + 396 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{x e + d} d^{3}\right )} A a b^{3} e^{\left (-3\right )} + 44 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{x e + d} d^{4}\right )} B a b^{3} e^{\left (-4\right )} + 11 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{x e + d} d^{4}\right )} A b^{4} e^{\left (-4\right )} + 5 \,{\left (63 \,{\left (x e + d\right )}^{\frac{11}{2}} - 385 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 990 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 1386 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} - 693 \, \sqrt{x e + d} d^{5}\right )} B b^{4} e^{\left (-5\right )} + 3465 \, \sqrt{x e + d} A a^{4}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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