Optimal. Leaf size=214 \[ -\frac{2 b^3 (d+e x)^{7/2} (-4 a B e-A b e+5 b B d)}{7 e^6}+\frac{4 b^2 (d+e x)^{5/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{5 e^6}-\frac{4 b (d+e x)^{3/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{3 e^6}+\frac{2 \sqrt{d+e x} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{e^6}+\frac{2 (b d-a e)^4 (B d-A e)}{e^6 \sqrt{d+e x}}+\frac{2 b^4 B (d+e x)^{9/2}}{9 e^6} \]
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Rubi [A] time = 0.0945112, antiderivative size = 214, 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} \[ -\frac{2 b^3 (d+e x)^{7/2} (-4 a B e-A b e+5 b B d)}{7 e^6}+\frac{4 b^2 (d+e x)^{5/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{5 e^6}-\frac{4 b (d+e x)^{3/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{3 e^6}+\frac{2 \sqrt{d+e x} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{e^6}+\frac{2 (b d-a e)^4 (B d-A e)}{e^6 \sqrt{d+e x}}+\frac{2 b^4 B (d+e x)^{9/2}}{9 e^6} \]
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}{(d+e x)^{3/2}} \, dx &=\int \frac{(a+b x)^4 (A+B x)}{(d+e x)^{3/2}} \, dx\\ &=\int \left (\frac{(-b d+a e)^4 (-B d+A e)}{e^5 (d+e x)^{3/2}}+\frac{(-b d+a e)^3 (-5 b B d+4 A b e+a B e)}{e^5 \sqrt{d+e x}}+\frac{2 b (b d-a e)^2 (-5 b B d+3 A b e+2 a B e) \sqrt{d+e x}}{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)^{3/2}}{e^5}+\frac{b^3 (-5 b B d+A b e+4 a B e) (d+e x)^{5/2}}{e^5}+\frac{b^4 B (d+e x)^{7/2}}{e^5}\right ) \, dx\\ &=\frac{2 (b d-a e)^4 (B d-A e)}{e^6 \sqrt{d+e x}}+\frac{2 (b d-a e)^3 (5 b B d-4 A b e-a B e) \sqrt{d+e x}}{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)^{3/2}}{3 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)^{5/2}}{5 e^6}-\frac{2 b^3 (5 b B d-A b e-4 a B e) (d+e x)^{7/2}}{7 e^6}+\frac{2 b^4 B (d+e x)^{9/2}}{9 e^6}\\ \end{align*}
Mathematica [A] time = 0.117795, size = 183, normalized size = 0.86 \[ \frac{2 \left (-45 b^3 (d+e x)^4 (-4 a B e-A b e+5 b B d)+126 b^2 (d+e x)^3 (b d-a e) (-3 a B e-2 A b e+5 b B d)-210 b (d+e x)^2 (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)+315 (d+e x) (b d-a e)^3 (-a B e-4 A b e+5 b B d)+315 (b d-a e)^4 (B d-A e)+35 b^4 B (d+e x)^5\right )}{315 e^6 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 469, normalized size = 2.2 \begin{align*} -{\frac{-70\,{b}^{4}B{x}^{5}{e}^{5}-90\,A{b}^{4}{e}^{5}{x}^{4}-360\,Ba{b}^{3}{e}^{5}{x}^{4}+100\,B{b}^{4}d{e}^{4}{x}^{4}-504\,Aa{b}^{3}{e}^{5}{x}^{3}+144\,A{b}^{4}d{e}^{4}{x}^{3}-756\,B{a}^{2}{b}^{2}{e}^{5}{x}^{3}+576\,Ba{b}^{3}d{e}^{4}{x}^{3}-160\,B{b}^{4}{d}^{2}{e}^{3}{x}^{3}-1260\,A{a}^{2}{b}^{2}{e}^{5}{x}^{2}+1008\,Aa{b}^{3}d{e}^{4}{x}^{2}-288\,A{b}^{4}{d}^{2}{e}^{3}{x}^{2}-840\,B{a}^{3}b{e}^{5}{x}^{2}+1512\,B{a}^{2}{b}^{2}d{e}^{4}{x}^{2}-1152\,Ba{b}^{3}{d}^{2}{e}^{3}{x}^{2}+320\,B{b}^{4}{d}^{3}{e}^{2}{x}^{2}-2520\,A{a}^{3}b{e}^{5}x+5040\,A{a}^{2}{b}^{2}d{e}^{4}x-4032\,Aa{b}^{3}{d}^{2}{e}^{3}x+1152\,A{b}^{4}{d}^{3}{e}^{2}x-630\,B{a}^{4}{e}^{5}x+3360\,B{a}^{3}bd{e}^{4}x-6048\,B{a}^{2}{b}^{2}{d}^{2}{e}^{3}x+4608\,Ba{b}^{3}{d}^{3}{e}^{2}x-1280\,B{b}^{4}{d}^{4}ex+630\,A{a}^{4}{e}^{5}-5040\,Ad{a}^{3}b{e}^{4}+10080\,A{d}^{2}{a}^{2}{b}^{2}{e}^{3}-8064\,Aa{b}^{3}{d}^{3}{e}^{2}+2304\,A{b}^{4}{d}^{4}e-1260\,B{a}^{4}d{e}^{4}+6720\,B{d}^{2}{a}^{3}b{e}^{3}-12096\,B{d}^{3}{a}^{2}{b}^{2}{e}^{2}+9216\,Ba{b}^{3}{d}^{4}e-2560\,{b}^{4}B{d}^{5}}{315\,{e}^{6}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.986727, size = 563, normalized size = 2.63 \begin{align*} \frac{2 \,{\left (\frac{35 \,{\left (e x + d\right )}^{\frac{9}{2}} B b^{4} - 45 \,{\left (5 \, B b^{4} d -{\left (4 \, B a b^{3} + A b^{4}\right )} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 126 \,{\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{5}{2}} - 210 \,{\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{3}{2}} + 315 \,{\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 )} \sqrt{e x + d}}{e^{5}} + \frac{315 \,{\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} e^{5}}\right )}}{315 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.33109, size = 919, normalized size = 4.29 \begin{align*} \frac{2 \,{\left (35 \, B b^{4} e^{5} x^{5} + 1280 \, B b^{4} d^{5} - 315 \, A a^{4} e^{5} - 1152 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 2016 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 1680 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 630 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} - 5 \,{\left (10 \, B b^{4} d e^{4} - 9 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 2 \,{\left (40 \, B b^{4} d^{2} e^{3} - 36 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 63 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} - 2 \,{\left (80 \, B b^{4} d^{3} e^{2} - 72 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 126 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} - 105 \,{\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 - 576 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 1008 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 840 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 315 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x\right )} \sqrt{e x + d}}{315 \,{\left (e^{7} x + d e^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 92.4987, size = 394, normalized size = 1.84 \begin{align*} \frac{2 B b^{4} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{6}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (2 A b^{4} e + 8 B a b^{3} e - 10 B b^{4} d\right )}{7 e^{6}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (8 A a b^{3} e^{2} - 8 A b^{4} d e + 12 B a^{2} b^{2} e^{2} - 32 B a b^{3} d e + 20 B b^{4} d^{2}\right )}{5 e^{6}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (12 A a^{2} b^{2} e^{3} - 24 A a b^{3} d e^{2} + 12 A b^{4} d^{2} e + 8 B a^{3} b e^{3} - 36 B a^{2} b^{2} d e^{2} + 48 B a b^{3} d^{2} e - 20 B b^{4} d^{3}\right )}{3 e^{6}} + \frac{\sqrt{d + e x} \left (8 A a^{3} b e^{4} - 24 A a^{2} b^{2} d e^{3} + 24 A a b^{3} d^{2} e^{2} - 8 A b^{4} d^{3} e + 2 B a^{4} e^{4} - 16 B a^{3} b d e^{3} + 36 B a^{2} b^{2} d^{2} e^{2} - 32 B a b^{3} d^{3} e + 10 B b^{4} d^{4}\right )}{e^{6}} + \frac{2 \left (- A e + B d\right ) \left (a e - b d\right )^{4}}{e^{6} \sqrt{d + e x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13671, size = 792, normalized size = 3.7 \begin{align*} \frac{2}{315} \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} B b^{4} e^{48} - 225 \,{\left (x e + d\right )}^{\frac{7}{2}} B b^{4} d e^{48} + 630 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{4} d^{2} e^{48} - 1050 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{4} d^{3} e^{48} + 1575 \, \sqrt{x e + d} B b^{4} d^{4} e^{48} + 180 \,{\left (x e + d\right )}^{\frac{7}{2}} B a b^{3} e^{49} + 45 \,{\left (x e + d\right )}^{\frac{7}{2}} A b^{4} e^{49} - 1008 \,{\left (x e + d\right )}^{\frac{5}{2}} B a b^{3} d e^{49} - 252 \,{\left (x e + d\right )}^{\frac{5}{2}} A b^{4} d e^{49} + 2520 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{3} d^{2} e^{49} + 630 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{4} d^{2} e^{49} - 5040 \, \sqrt{x e + d} B a b^{3} d^{3} e^{49} - 1260 \, \sqrt{x e + d} A b^{4} d^{3} e^{49} + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} B a^{2} b^{2} e^{50} + 252 \,{\left (x e + d\right )}^{\frac{5}{2}} A a b^{3} e^{50} - 1890 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b^{2} d e^{50} - 1260 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{3} d e^{50} + 5670 \, \sqrt{x e + d} B a^{2} b^{2} d^{2} e^{50} + 3780 \, \sqrt{x e + d} A a b^{3} d^{2} e^{50} + 420 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{3} b e^{51} + 630 \,{\left (x e + d\right )}^{\frac{3}{2}} A a^{2} b^{2} e^{51} - 2520 \, \sqrt{x e + d} B a^{3} b d e^{51} - 3780 \, \sqrt{x e + d} A a^{2} b^{2} d e^{51} + 315 \, \sqrt{x e + d} B a^{4} e^{52} + 1260 \, \sqrt{x e + d} A a^{3} b e^{52}\right )} e^{\left (-54\right )} + \frac{2 \,{\left (B b^{4} d^{5} - 4 \, B a b^{3} d^{4} e - A b^{4} d^{4} e + 6 \, B a^{2} b^{2} d^{3} e^{2} + 4 \, A a b^{3} d^{3} e^{2} - 4 \, B a^{3} b d^{2} e^{3} - 6 \, A a^{2} b^{2} d^{2} e^{3} + B a^{4} d e^{4} + 4 \, A a^{3} b d e^{4} - A a^{4} e^{5}\right )} e^{\left (-6\right )}}{\sqrt{x e + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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