Optimal. Leaf size=112 \[ \frac {2 \sqrt {d+e x} \left (a B e^2-2 A c d e+3 B c d^2\right )}{e^4}+\frac {2 \left (a e^2+c d^2\right ) (B d-A e)}{e^4 \sqrt {d+e x}}-\frac {2 c (d+e x)^{3/2} (3 B d-A e)}{3 e^4}+\frac {2 B c (d+e x)^{5/2}}{5 e^4} \]
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Rubi [A] time = 0.05, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {772} \begin {gather*} \frac {2 \sqrt {d+e x} \left (a B e^2-2 A c d e+3 B c d^2\right )}{e^4}+\frac {2 \left (a e^2+c d^2\right ) (B d-A e)}{e^4 \sqrt {d+e x}}-\frac {2 c (d+e x)^{3/2} (3 B d-A e)}{3 e^4}+\frac {2 B c (d+e x)^{5/2}}{5 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 772
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )}{(d+e x)^{3/2}} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right )}{e^3 (d+e x)^{3/2}}+\frac {3 B c d^2-2 A c d e+a B e^2}{e^3 \sqrt {d+e x}}+\frac {c (-3 B d+A e) \sqrt {d+e x}}{e^3}+\frac {B c (d+e x)^{3/2}}{e^3}\right ) \, dx\\ &=\frac {2 (B d-A e) \left (c d^2+a e^2\right )}{e^4 \sqrt {d+e x}}+\frac {2 \left (3 B c d^2-2 A c d e+a B e^2\right ) \sqrt {d+e x}}{e^4}-\frac {2 c (3 B d-A e) (d+e x)^{3/2}}{3 e^4}+\frac {2 B c (d+e x)^{5/2}}{5 e^4}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 97, normalized size = 0.87 \begin {gather*} \frac {6 B \left (5 a e^2 (2 d+e x)+c \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )-10 A e \left (3 a e^2+c \left (8 d^2+4 d e x-e^2 x^2\right )\right )}{15 e^4 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 117, normalized size = 1.04 \begin {gather*} \frac {2 \left (-15 a A e^3+15 a B e^2 (d+e x)+15 a B d e^2-15 A c d^2 e-30 A c d e (d+e x)+5 A c e (d+e x)^2+15 B c d^3+45 B c d^2 (d+e x)-15 B c d (d+e x)^2+3 B c (d+e x)^3\right )}{15 e^4 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 110, normalized size = 0.98 \begin {gather*} \frac {2 \, {\left (3 \, B c e^{3} x^{3} + 48 \, B c d^{3} - 40 \, A c d^{2} e + 30 \, B a d e^{2} - 15 \, A a e^{3} - {\left (6 \, B c d e^{2} - 5 \, A c e^{3}\right )} x^{2} + {\left (24 \, B c d^{2} e - 20 \, A c d e^{2} + 15 \, B a e^{3}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{5} x + d e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 135, normalized size = 1.21 \begin {gather*} \frac {2}{15} \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} B c e^{16} - 15 \, {\left (x e + d\right )}^{\frac {3}{2}} B c d e^{16} + 45 \, \sqrt {x e + d} B c d^{2} e^{16} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} A c e^{17} - 30 \, \sqrt {x e + d} A c d e^{17} + 15 \, \sqrt {x e + d} B a e^{18}\right )} e^{\left (-20\right )} + \frac {2 \, {\left (B c d^{3} - A c d^{2} e + B a d e^{2} - A a e^{3}\right )} e^{\left (-4\right )}}{\sqrt {x e + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 101, normalized size = 0.90 \begin {gather*} -\frac {2 \left (-3 B c \,x^{3} e^{3}-5 A c \,e^{3} x^{2}+6 B c d \,e^{2} x^{2}+20 A c d \,e^{2} x -15 B a \,e^{3} x -24 B c \,d^{2} e x +15 a A \,e^{3}+40 A c \,d^{2} e -30 a B d \,e^{2}-48 B c \,d^{3}\right )}{15 \sqrt {e x +d}\, e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 112, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (\frac {3 \, {\left (e x + d\right )}^{\frac {5}{2}} B c - 5 \, {\left (3 \, B c d - A c e\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 15 \, {\left (3 \, B c d^{2} - 2 \, A c d e + B a e^{2}\right )} \sqrt {e x + d}}{e^{3}} + \frac {15 \, {\left (B c d^{3} - A c d^{2} e + B a d e^{2} - A a e^{3}\right )}}{\sqrt {e x + d} e^{3}}\right )}}{15 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 111, normalized size = 0.99 \begin {gather*} \frac {\sqrt {d+e\,x}\,\left (6\,B\,c\,d^2-4\,A\,c\,d\,e+2\,B\,a\,e^2\right )}{e^4}-\frac {-2\,B\,c\,d^3+2\,A\,c\,d^2\,e-2\,B\,a\,d\,e^2+2\,A\,a\,e^3}{e^4\,\sqrt {d+e\,x}}+\frac {2\,B\,c\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4}+\frac {2\,c\,\left (A\,e-3\,B\,d\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 21.21, size = 112, normalized size = 1.00 \begin {gather*} \frac {2 B c \left (d + e x\right )^{\frac {5}{2}}}{5 e^{4}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (2 A c e - 6 B c d\right )}{3 e^{4}} + \frac {\sqrt {d + e x} \left (- 4 A c d e + 2 B a e^{2} + 6 B c d^{2}\right )}{e^{4}} + \frac {2 \left (- A e + B d\right ) \left (a e^{2} + c d^{2}\right )}{e^{4} \sqrt {d + e x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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