Optimal. Leaf size=126 \[ -\frac{2 b (d+e x)^{5/2} (-2 a B e-A b e+3 b B d)}{5 e^4}+\frac{2 (d+e x)^{3/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{3 e^4}-\frac{2 \sqrt{d+e x} (b d-a e)^2 (B d-A e)}{e^4}+\frac{2 b^2 B (d+e x)^{7/2}}{7 e^4} \]
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Rubi [A] time = 0.0524879, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 77} \[ -\frac{2 b (d+e x)^{5/2} (-2 a B e-A b e+3 b B d)}{5 e^4}+\frac{2 (d+e x)^{3/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{3 e^4}-\frac{2 \sqrt{d+e x} (b d-a e)^2 (B d-A e)}{e^4}+\frac{2 b^2 B (d+e x)^{7/2}}{7 e^4} \]
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 )}{\sqrt{d+e x}} \, dx &=\int \frac{(a+b x)^2 (A+B x)}{\sqrt{d+e x}} \, dx\\ &=\int \left (\frac{(-b d+a e)^2 (-B d+A e)}{e^3 \sqrt{d+e x}}+\frac{(-b d+a e) (-3 b B d+2 A b e+a B e) \sqrt{d+e x}}{e^3}+\frac{b (-3 b B d+A b e+2 a B e) (d+e x)^{3/2}}{e^3}+\frac{b^2 B (d+e x)^{5/2}}{e^3}\right ) \, dx\\ &=-\frac{2 (b d-a e)^2 (B d-A e) \sqrt{d+e x}}{e^4}+\frac{2 (b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{3/2}}{3 e^4}-\frac{2 b (3 b B d-A b e-2 a B e) (d+e x)^{5/2}}{5 e^4}+\frac{2 b^2 B (d+e x)^{7/2}}{7 e^4}\\ \end{align*}
Mathematica [A] time = 0.0924312, size = 107, normalized size = 0.85 \[ \frac{2 \sqrt{d+e x} \left (-21 b (d+e x)^2 (-2 a B e-A b e+3 b B d)+35 (d+e x) (b d-a e) (-a B e-2 A b e+3 b B d)-105 (b d-a e)^2 (B d-A e)+15 b^2 B (d+e x)^3\right )}{105 e^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 169, normalized size = 1.3 \begin{align*}{\frac{30\,{b}^{2}B{x}^{3}{e}^{3}+42\,A{b}^{2}{e}^{3}{x}^{2}+84\,Bab{e}^{3}{x}^{2}-36\,B{b}^{2}d{e}^{2}{x}^{2}+140\,Axab{e}^{3}-56\,Ax{b}^{2}d{e}^{2}+70\,Bx{a}^{2}{e}^{3}-112\,Bxabd{e}^{2}+48\,B{b}^{2}{d}^{2}ex+210\,A{a}^{2}{e}^{3}-280\,Aabd{e}^{2}+112\,A{b}^{2}{d}^{2}e-140\,B{a}^{2}d{e}^{2}+224\,Bab{d}^{2}e-96\,{b}^{2}B{d}^{3}}{105\,{e}^{4}}\sqrt{ex+d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.988725, size = 215, normalized size = 1.71 \begin{align*} \frac{2 \,{\left (15 \,{\left (e x + d\right )}^{\frac{7}{2}} B b^{2} - 21 \,{\left (3 \, B b^{2} d -{\left (2 \, B a b + A b^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (3 \, B b^{2} d^{2} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d e +{\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 105 \,{\left (B b^{2} d^{3} - A a^{2} e^{3} -{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )} \sqrt{e x + d}\right )}}{105 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36742, size = 351, normalized size = 2.79 \begin{align*} \frac{2 \,{\left (15 \, B b^{2} e^{3} x^{3} - 48 \, B b^{2} d^{3} + 105 \, A a^{2} e^{3} + 56 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e - 70 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} - 3 \,{\left (6 \, B b^{2} d e^{2} - 7 \,{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} +{\left (24 \, B b^{2} d^{2} e - 28 \,{\left (2 \, B a b + A b^{2}\right )} d e^{2} + 35 \,{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 44.7561, size = 583, normalized size = 4.63 \begin{align*} \begin{cases} - \frac{\frac{2 A a^{2} d}{\sqrt{d + e x}} + 2 A a^{2} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{4 A a b d \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{4 A a b \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e} + \frac{2 A b^{2} d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{2 A b^{2} \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 B a^{2} d \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{2 B a^{2} \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e} + \frac{4 B a b d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{4 B a b \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 B b^{2} d \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{3}} + \frac{2 B b^{2} \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}}}{e} & \text{for}\: e \neq 0 \\\frac{A a^{2} x + \frac{B b^{2} x^{4}}{4} + \frac{x^{3} \left (A b^{2} + 2 B a b\right )}{3} + \frac{x^{2} \left (2 A a b + B a^{2}\right )}{2}}{\sqrt{d}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13638, size = 290, normalized size = 2.3 \begin{align*} \frac{2}{105} \,{\left (35 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} B a^{2} e^{\left (-1\right )} + 70 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} A a b e^{\left (-1\right )} + 14 \,{\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 b e^{\left (-2\right )} + 7 \,{\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 b^{2} e^{\left (-2\right )} + 3 \,{\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 b^{2} e^{\left (-3\right )} + 105 \, \sqrt{x e + d} A a^{2}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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