Optimal. Leaf size=122 \[ \frac {2 (-A c e-b B e+3 B c d)}{e^4 \sqrt {d+e x}}-\frac {2 (B d (3 c d-2 b e)-A e (2 c d-b e))}{3 e^4 (d+e x)^{3/2}}+\frac {2 d (B d-A e) (c d-b e)}{5 e^4 (d+e x)^{5/2}}+\frac {2 B c \sqrt {d+e x}}{e^4} \]
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Rubi [A] time = 0.07, 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 (-A c e-b B e+3 B c d)}{e^4 \sqrt {d+e x}}-\frac {2 (B d (3 c d-2 b e)-A e (2 c d-b e))}{3 e^4 (d+e x)^{3/2}}+\frac {2 d (B d-A e) (c d-b e)}{5 e^4 (d+e x)^{5/2}}+\frac {2 B c \sqrt {d+e x}}{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)^{7/2}} \, dx &=\int \left (-\frac {d (B d-A e) (c d-b e)}{e^3 (d+e x)^{7/2}}+\frac {B d (3 c d-2 b e)-A e (2 c d-b e)}{e^3 (d+e x)^{5/2}}+\frac {-3 B c d+b B e+A c e}{e^3 (d+e x)^{3/2}}+\frac {B c}{e^3 \sqrt {d+e x}}\right ) \, dx\\ &=\frac {2 d (B d-A e) (c d-b e)}{5 e^4 (d+e x)^{5/2}}-\frac {2 (B d (3 c d-2 b e)-A e (2 c d-b e))}{3 e^4 (d+e x)^{3/2}}+\frac {2 (3 B c d-b B e-A c e)}{e^4 \sqrt {d+e x}}+\frac {2 B c \sqrt {d+e x}}{e^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 111, normalized size = 0.91 \begin {gather*} -\frac {2 \left (A e \left (b e (2 d+5 e x)+c \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )+B \left (b e \left (8 d^2+20 d e x+15 e^2 x^2\right )-3 c \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )\right )\right )}{15 e^4 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 141, normalized size = 1.16 \begin {gather*} \frac {2 \left (-5 A b e^2 (d+e x)+3 A b d e^2-3 A c d^2 e+10 A c d e (d+e x)-15 A c e (d+e x)^2-3 b B d^2 e+10 b B d e (d+e x)-15 b B e (d+e x)^2+3 B c d^3-15 B c d^2 (d+e x)+45 B c d (d+e x)^2+15 B c (d+e x)^3\right )}{15 e^4 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 141, normalized size = 1.16 \begin {gather*} \frac {2 \, {\left (15 \, B c e^{3} x^{3} + 48 \, B c d^{3} - 2 \, A b d e^{2} - 8 \, {\left (B b + A c\right )} d^{2} e + 15 \, {\left (6 \, B c d e^{2} - {\left (B b + A c\right )} e^{3}\right )} x^{2} + 5 \, {\left (24 \, B c d^{2} e - A b e^{3} - 4 \, {\left (B b + A c\right )} d e^{2}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 152, normalized size = 1.25 \begin {gather*} 2 \, \sqrt {x e + d} B c e^{\left (-4\right )} + \frac {2 \, {\left (45 \, {\left (x e + d\right )}^{2} B c d - 15 \, {\left (x e + d\right )} B c d^{2} + 3 \, B c d^{3} - 15 \, {\left (x e + d\right )}^{2} B b e - 15 \, {\left (x e + d\right )}^{2} A c e + 10 \, {\left (x e + d\right )} B b d e + 10 \, {\left (x e + d\right )} A c d e - 3 \, B b d^{2} e - 3 \, A c d^{2} e - 5 \, {\left (x e + d\right )} A b e^{2} + 3 \, A b d e^{2}\right )} e^{\left (-4\right )}}{15 \, {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 121, normalized size = 0.99 \begin {gather*} -\frac {2 \left (-15 B c \,x^{3} e^{3}+15 A c \,e^{3} x^{2}+15 B b \,e^{3} x^{2}-90 B c d \,e^{2} x^{2}+5 A b \,e^{3} x +20 A c d \,e^{2} x +20 B b d \,e^{2} x -120 B c \,d^{2} e x +2 A b d \,e^{2}+8 A c \,d^{2} e +8 B b \,d^{2} e -48 B c \,d^{3}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 117, normalized size = 0.96 \begin {gather*} \frac {2 \, {\left (\frac {15 \, \sqrt {e x + d} B c}{e^{3}} + \frac {3 \, B c d^{3} + 3 \, A b d e^{2} - 3 \, {\left (B b + A c\right )} d^{2} e + 15 \, {\left (3 \, B c d - {\left (B b + A c\right )} e\right )} {\left (e x + d\right )}^{2} - 5 \, {\left (3 \, B c d^{2} + A b e^{2} - 2 \, {\left (B b + A c\right )} d e\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} e^{3}}\right )}}{15 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 120, normalized size = 0.98 \begin {gather*} -\frac {2\,\left (2\,A\,b\,d\,e^2-48\,B\,c\,d^3+8\,A\,c\,d^2\,e+8\,B\,b\,d^2\,e+5\,A\,b\,e^3\,x+15\,A\,c\,e^3\,x^2+15\,B\,b\,e^3\,x^2-15\,B\,c\,e^3\,x^3-90\,B\,c\,d\,e^2\,x^2+20\,A\,c\,d\,e^2\,x+20\,B\,b\,d\,e^2\,x-120\,B\,c\,d^2\,e\,x\right )}{15\,e^4\,{\left (d+e\,x\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.43, size = 784, normalized size = 6.43 \begin {gather*} \begin {cases} - \frac {4 A b d e^{2}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {10 A b e^{3} x}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {16 A c d^{2} e}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {40 A c d e^{2} x}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {30 A c e^{3} x^{2}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {16 B b d^{2} e}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {40 B b d e^{2} x}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {30 B b e^{3} x^{2}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} + \frac {96 B c d^{3}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} + \frac {240 B c d^{2} e x}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} + \frac {180 B c d e^{2} x^{2}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} + \frac {30 B c e^{3} x^{3}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {\frac {A b x^{2}}{2} + \frac {A c x^{3}}{3} + \frac {B b x^{3}}{3} + \frac {B c x^{4}}{4}}{d^{\frac {7}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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