Optimal. Leaf size=122 \[ -\frac {2 (d+e x)^{3/2} (-A c e-b B e+3 B c d)}{3 e^4}+\frac {2 \sqrt {d+e x} (B d (3 c d-2 b e)-A e (2 c d-b e))}{e^4}+\frac {2 d (B d-A e) (c d-b e)}{e^4 \sqrt {d+e x}}+\frac {2 B c (d+e x)^{5/2}}{5 e^4} \]
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Rubi [A] time = 0.08, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \begin {gather*} -\frac {2 (d+e x)^{3/2} (-A c e-b B e+3 B c d)}{3 e^4}+\frac {2 \sqrt {d+e x} (B d (3 c d-2 b e)-A e (2 c d-b e))}{e^4}+\frac {2 d (B d-A e) (c d-b e)}{e^4 \sqrt {d+e x}}+\frac {2 B c (d+e x)^{5/2}}{5 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )}{(d+e x)^{3/2}} \, dx &=\int \left (-\frac {d (B d-A e) (c d-b e)}{e^3 (d+e x)^{3/2}}+\frac {B d (3 c d-2 b e)-A e (2 c d-b e)}{e^3 \sqrt {d+e x}}+\frac {(-3 B c d+b B e+A c e) \sqrt {d+e x}}{e^3}+\frac {B c (d+e x)^{3/2}}{e^3}\right ) \, dx\\ &=\frac {2 d (B d-A e) (c d-b e)}{e^4 \sqrt {d+e x}}+\frac {2 (B d (3 c d-2 b e)-A e (2 c d-b e)) \sqrt {d+e x}}{e^4}-\frac {2 (3 B c d-b B e-A c 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 = 0.05, size = 110, normalized size = 0.90 \begin {gather*} \frac {2 \left (5 A e \left (3 b e (2 d+e x)+c \left (-8 d^2-4 d e x+e^2 x^2\right )\right )+B \left (5 b e \left (-8 d^2-4 d e x+e^2 x^2\right )+3 c \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )\right )}{15 e^4 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 141, normalized size = 1.16 \begin {gather*} \frac {2 \left (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 B d^2 e-30 b B d e (d+e x)+5 b B 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 = 0.41, size = 118, normalized size = 0.97 \begin {gather*} \frac {2 \, {\left (3 \, B c e^{3} x^{3} + 48 \, B c d^{3} + 30 \, A b d e^{2} - 40 \, {\left (B b + A c\right )} d^{2} e - {\left (6 \, B c d e^{2} - 5 \, {\left (B b + A c\right )} e^{3}\right )} x^{2} + {\left (24 \, B c d^{2} e + 15 \, A b e^{3} - 20 \, {\left (B b + A c\right )} d e^{2}\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 = 0.18, size = 167, normalized size = 1.37 \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}} B b e^{17} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} A c e^{17} - 30 \, \sqrt {x e + d} B b d e^{17} - 30 \, \sqrt {x e + d} A c d e^{17} + 15 \, \sqrt {x e + d} A b e^{18}\right )} e^{\left (-20\right )} + \frac {2 \, {\left (B c d^{3} - B b d^{2} e - A c d^{2} e + A b d e^{2}\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 = 0.06, size = 121, normalized size = 0.99 \begin {gather*} \frac {\frac {2}{5} B c \,x^{3} e^{3}+\frac {2}{3} A c \,e^{3} x^{2}+\frac {2}{3} B b \,e^{3} x^{2}-\frac {4}{5} B c d \,e^{2} x^{2}+2 A b \,e^{3} x -\frac {8}{3} A c d \,e^{2} x -\frac {8}{3} B b d \,e^{2} x +\frac {16}{5} B c \,d^{2} e x +4 A b d \,e^{2}-\frac {16}{3} A c \,d^{2} e -\frac {16}{3} B b \,d^{2} e +\frac {32}{5} B c \,d^{3}}{\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 = 0.57, size = 120, normalized size = 0.98 \begin {gather*} \frac {2 \, {\left (\frac {3 \, {\left (e x + d\right )}^{\frac {5}{2}} B c - 5 \, {\left (3 \, B c d - {\left (B b + A c\right )} e\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 15 \, {\left (3 \, B c d^{2} + A b e^{2} - 2 \, {\left (B b + A c\right )} d e\right )} \sqrt {e x + d}}{e^{3}} + \frac {15 \, {\left (B c d^{3} + A b d e^{2} - {\left (B b + A c\right )} d^{2} e\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 = 1.55, size = 124, normalized size = 1.02 \begin {gather*} \frac {\sqrt {d+e\,x}\,\left (2\,A\,b\,e^2+6\,B\,c\,d^2-4\,A\,c\,d\,e-4\,B\,b\,d\,e\right )}{e^4}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (2\,A\,c\,e+2\,B\,b\,e-6\,B\,c\,d\right )}{3\,e^4}+\frac {2\,B\,c\,d^3+2\,A\,b\,d\,e^2-2\,A\,c\,d^2\,e-2\,B\,b\,d^2\,e}{e^4\,\sqrt {d+e\,x}}+\frac {2\,B\,c\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 24.16, size = 126, normalized size = 1.03 \begin {gather*} \frac {2 B c \left (d + e x\right )^{\frac {5}{2}}}{5 e^{4}} - \frac {2 d \left (- A e + B d\right ) \left (b e - c d\right )}{e^{4} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (2 A c e + 2 B b e - 6 B c d\right )}{3 e^{4}} + \frac {\sqrt {d + e x} \left (2 A b e^{2} - 4 A c d e - 4 B b d e + 6 B c d^{2}\right )}{e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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