Optimal. Leaf size=124 \[ -\frac{2 b (d+e x)^{3/2} (-2 a B e-A b e+3 b B d)}{3 e^4}+\frac{2 \sqrt{d+e x} (b d-a e) (-a B e-2 A b e+3 b B d)}{e^4}+\frac{2 (b d-a e)^2 (B d-A e)}{e^4 \sqrt{d+e x}}+\frac{2 b^2 B (d+e x)^{5/2}}{5 e^4} \]
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Rubi [A] time = 0.0546105, antiderivative size = 124, 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} \[ -\frac{2 b (d+e x)^{3/2} (-2 a B e-A b e+3 b B d)}{3 e^4}+\frac{2 \sqrt{d+e x} (b d-a e) (-a B e-2 A b e+3 b B d)}{e^4}+\frac{2 (b d-a e)^2 (B d-A e)}{e^4 \sqrt{d+e x}}+\frac{2 b^2 B (d+e x)^{5/2}}{5 e^4} \]
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 )}{(d+e x)^{3/2}} \, dx &=\int \frac{(a+b x)^2 (A+B x)}{(d+e x)^{3/2}} \, dx\\ &=\int \left (\frac{(-b d+a e)^2 (-B d+A e)}{e^3 (d+e x)^{3/2}}+\frac{(-b d+a e) (-3 b B d+2 A b e+a B e)}{e^3 \sqrt{d+e x}}+\frac{b (-3 b B d+A b e+2 a B e) \sqrt{d+e x}}{e^3}+\frac{b^2 B (d+e x)^{3/2}}{e^3}\right ) \, dx\\ &=\frac{2 (b d-a e)^2 (B d-A e)}{e^4 \sqrt{d+e x}}+\frac{2 (b d-a e) (3 b B d-2 A b e-a B e) \sqrt{d+e x}}{e^4}-\frac{2 b (3 b B d-A b e-2 a B e) (d+e x)^{3/2}}{3 e^4}+\frac{2 b^2 B (d+e x)^{5/2}}{5 e^4}\\ \end{align*}
Mathematica [A] time = 0.0919046, size = 107, normalized size = 0.86 \[ \frac{2 \left (-5 b (d+e x)^2 (-2 a B e-A b e+3 b B d)+15 (d+e x) (b d-a e) (-a B e-2 A b e+3 b B d)+15 (b d-a e)^2 (B d-A e)+3 b^2 B (d+e x)^3\right )}{15 e^4 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 169, normalized size = 1.4 \begin{align*} -{\frac{-6\,{b}^{2}B{x}^{3}{e}^{3}-10\,A{b}^{2}{e}^{3}{x}^{2}-20\,Bab{e}^{3}{x}^{2}+12\,B{b}^{2}d{e}^{2}{x}^{2}-60\,Axab{e}^{3}+40\,Ax{b}^{2}d{e}^{2}-30\,Bx{a}^{2}{e}^{3}+80\,Bxabd{e}^{2}-48\,B{b}^{2}{d}^{2}ex+30\,A{a}^{2}{e}^{3}-120\,Aabd{e}^{2}+80\,A{b}^{2}{d}^{2}e-60\,B{a}^{2}d{e}^{2}+160\,Bab{d}^{2}e-96\,{b}^{2}B{d}^{3}}{15\,{e}^{4}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00766, size = 225, normalized size = 1.81 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (e x + d\right )}^{\frac{5}{2}} B b^{2} - 5 \,{\left (3 \, B b^{2} d -{\left (2 \, B a b + A b^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 15 \,{\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 )} \sqrt{e x + d}}{e^{3}} + \frac{15 \,{\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} e^{3}}\right )}}{15 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28246, size = 360, normalized size = 2.9 \begin{align*} \frac{2 \,{\left (3 \, B b^{2} e^{3} x^{3} + 48 \, B b^{2} d^{3} - 15 \, A a^{2} e^{3} - 40 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e + 30 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} -{\left (6 \, B b^{2} d e^{2} - 5 \,{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} +{\left (24 \, B b^{2} d^{2} e - 20 \,{\left (2 \, B a b + A b^{2}\right )} d e^{2} + 15 \,{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{5} x + d e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 26.1372, size = 150, normalized size = 1.21 \begin{align*} \frac{2 B b^{2} \left (d + e x\right )^{\frac{5}{2}}}{5 e^{4}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (2 A b^{2} e + 4 B a b e - 6 B b^{2} d\right )}{3 e^{4}} + \frac{\sqrt{d + e x} \left (4 A a b e^{2} - 4 A b^{2} d e + 2 B a^{2} e^{2} - 8 B a b d e + 6 B b^{2} d^{2}\right )}{e^{4}} + \frac{2 \left (- A e + B d\right ) \left (a e - b d\right )^{2}}{e^{4} \sqrt{d + e x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12891, size = 296, normalized size = 2.39 \begin{align*} \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{2} e^{16} - 15 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} d e^{16} + 45 \, \sqrt{x e + d} B b^{2} d^{2} e^{16} + 10 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b e^{17} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{2} e^{17} - 60 \, \sqrt{x e + d} B a b d e^{17} - 30 \, \sqrt{x e + d} A b^{2} d e^{17} + 15 \, \sqrt{x e + d} B a^{2} e^{18} + 30 \, \sqrt{x e + d} A a b e^{18}\right )} e^{\left (-20\right )} + \frac{2 \,{\left (B b^{2} d^{3} - 2 \, B a b d^{2} e - A b^{2} d^{2} e + B a^{2} d e^{2} + 2 \, A a b d e^{2} - A a^{2} e^{3}\right )} e^{\left (-4\right )}}{\sqrt{x e + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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