Optimal. Leaf size=263 \[ -\frac {2 (d+e x)^{3/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{3 e^6}+\frac {2 \sqrt {d+e x} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^6}+\frac {2 d^2 (B d-A e) (c d-b e)^2}{3 e^6 (d+e x)^{3/2}}-\frac {2 c (d+e x)^{5/2} (-A c e-2 b B e+5 B c d)}{5 e^6}-\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 \sqrt {d+e x}}+\frac {2 B c^2 (d+e x)^{7/2}}{7 e^6} \]
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Rubi [A] time = 0.15, antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {771} \begin {gather*} -\frac {2 (d+e x)^{3/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{3 e^6}+\frac {2 \sqrt {d+e x} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^6}+\frac {2 d^2 (B d-A e) (c d-b e)^2}{3 e^6 (d+e x)^{3/2}}-\frac {2 c (d+e x)^{5/2} (-A c e-2 b B e+5 B c d)}{5 e^6}-\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 \sqrt {d+e x}}+\frac {2 B c^2 (d+e x)^{7/2}}{7 e^6} \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 )^2}{(d+e x)^{5/2}} \, dx &=\int \left (-\frac {d^2 (B d-A e) (c d-b e)^2}{e^5 (d+e x)^{5/2}}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^5 (d+e x)^{3/2}}+\frac {A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )}{e^5 \sqrt {d+e x}}+\frac {\left (-2 A c e (2 c d-b e)+B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) \sqrt {d+e x}}{e^5}+\frac {c (-5 B c d+2 b B e+A c e) (d+e x)^{3/2}}{e^5}+\frac {B c^2 (d+e x)^{5/2}}{e^5}\right ) \, dx\\ &=\frac {2 d^2 (B d-A e) (c d-b e)^2}{3 e^6 (d+e x)^{3/2}}-\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 \sqrt {d+e x}}+\frac {2 \left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) \sqrt {d+e x}}{e^6}-\frac {2 \left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{3/2}}{3 e^6}-\frac {2 c (5 B c d-2 b B e-A c e) (d+e x)^{5/2}}{5 e^6}+\frac {2 B c^2 (d+e x)^{7/2}}{7 e^6}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 271, normalized size = 1.03 \begin {gather*} \frac {2 \left (7 A e \left (5 b^2 e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+10 b c e \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+c^2 \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )\right )+B \left (35 b^2 e^2 \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+14 b c e \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )-5 c^2 \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )\right )\right )}{105 e^6 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 399, normalized size = 1.52 \begin {gather*} \frac {2 \left (-35 A b^2 d^2 e^3+210 A b^2 d e^3 (d+e x)+105 A b^2 e^3 (d+e x)^2+70 A b c d^3 e^2-630 A b c d^2 e^2 (d+e x)-630 A b c d e^2 (d+e x)^2+70 A b c e^2 (d+e x)^3-35 A c^2 d^4 e+420 A c^2 d^3 e (d+e x)+630 A c^2 d^2 e (d+e x)^2-140 A c^2 d e (d+e x)^3+21 A c^2 e (d+e x)^4+35 b^2 B d^3 e^2-315 b^2 B d^2 e^2 (d+e x)-315 b^2 B d e^2 (d+e x)^2+35 b^2 B e^2 (d+e x)^3-70 b B c d^4 e+840 b B c d^3 e (d+e x)+1260 b B c d^2 e (d+e x)^2-280 b B c d e (d+e x)^3+42 b B c e (d+e x)^4+35 B c^2 d^5-525 B c^2 d^4 (d+e x)-1050 B c^2 d^3 (d+e x)^2+350 B c^2 d^2 (d+e x)^3-105 B c^2 d (d+e x)^4+15 B c^2 (d+e x)^5\right )}{105 e^6 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 310, normalized size = 1.18 \begin {gather*} \frac {2 \, {\left (15 \, B c^{2} e^{5} x^{5} - 1280 \, B c^{2} d^{5} + 280 \, A b^{2} d^{2} e^{3} + 896 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e - 560 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - 3 \, {\left (10 \, B c^{2} d e^{4} - 7 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + {\left (80 \, B c^{2} d^{2} e^{3} - 56 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 35 \, {\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} - 3 \, {\left (160 \, B c^{2} d^{3} e^{2} - 35 \, A b^{2} e^{5} - 112 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 70 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} - 12 \, {\left (160 \, B c^{2} d^{4} e - 35 \, A b^{2} d e^{4} - 112 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 70 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{105 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 427, normalized size = 1.62 \begin {gather*} \frac {2}{105} \, {\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} B c^{2} e^{36} - 105 \, {\left (x e + d\right )}^{\frac {5}{2}} B c^{2} d e^{36} + 350 \, {\left (x e + d\right )}^{\frac {3}{2}} B c^{2} d^{2} e^{36} - 1050 \, \sqrt {x e + d} B c^{2} d^{3} e^{36} + 42 \, {\left (x e + d\right )}^{\frac {5}{2}} B b c e^{37} + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} A c^{2} e^{37} - 280 \, {\left (x e + d\right )}^{\frac {3}{2}} B b c d e^{37} - 140 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{2} d e^{37} + 1260 \, \sqrt {x e + d} B b c d^{2} e^{37} + 630 \, \sqrt {x e + d} A c^{2} d^{2} e^{37} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{2} e^{38} + 70 \, {\left (x e + d\right )}^{\frac {3}{2}} A b c e^{38} - 315 \, \sqrt {x e + d} B b^{2} d e^{38} - 630 \, \sqrt {x e + d} A b c d e^{38} + 105 \, \sqrt {x e + d} A b^{2} e^{39}\right )} e^{\left (-42\right )} - \frac {2 \, {\left (15 \, {\left (x e + d\right )} B c^{2} d^{4} - B c^{2} d^{5} - 24 \, {\left (x e + d\right )} B b c d^{3} e - 12 \, {\left (x e + d\right )} A c^{2} d^{3} e + 2 \, B b c d^{4} e + A c^{2} d^{4} e + 9 \, {\left (x e + d\right )} B b^{2} d^{2} e^{2} + 18 \, {\left (x e + d\right )} A b c d^{2} e^{2} - B b^{2} d^{3} e^{2} - 2 \, A b c d^{3} e^{2} - 6 \, {\left (x e + d\right )} A b^{2} d e^{3} + A b^{2} d^{2} e^{3}\right )} e^{\left (-6\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 341, normalized size = 1.30 \begin {gather*} \frac {\frac {512}{15} B b c \,d^{4} e -\frac {4}{7} B \,c^{2} d \,e^{4} x^{4}+2 A \,b^{2} e^{5} x^{2}+\frac {16}{3} A \,b^{2} d^{2} e^{3}+\frac {256}{15} A \,c^{2} d^{4} e -\frac {32}{3} B \,b^{2} d^{3} e^{2}+\frac {2}{3} B \,b^{2} e^{5} x^{3}+\frac {2}{5} A \,c^{2} e^{5} x^{4}+\frac {256}{5} B b c \,d^{3} e^{2} x -32 A b c \,d^{2} e^{3} x -8 A b c d \,e^{4} x^{2}+\frac {64}{5} B b c \,d^{2} e^{3} x^{2}-\frac {32}{15} B b c d \,e^{4} x^{3}+\frac {2}{7} B \,c^{2} x^{5} e^{5}-\frac {512}{21} B \,c^{2} d^{5}+\frac {32}{21} B \,c^{2} d^{2} e^{3} x^{3}+\frac {128}{5} A \,c^{2} d^{3} e^{2} x -16 B \,b^{2} d^{2} e^{3} x -\frac {16}{15} A \,c^{2} d \,e^{4} x^{3}+\frac {32}{5} A \,c^{2} d^{2} e^{3} x^{2}-4 B \,b^{2} d \,e^{4} x^{2}-\frac {64}{7} B \,c^{2} d^{3} e^{2} x^{2}+8 A \,b^{2} d \,e^{4} x +\frac {4}{5} B b c \,e^{5} x^{4}+\frac {4}{3} A b c \,e^{5} x^{3}-\frac {64}{3} A b c \,d^{3} e^{2}-\frac {256}{7} B \,c^{2} d^{4} e x}{\left (e x +d \right )^{\frac {3}{2}} e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 297, normalized size = 1.13 \begin {gather*} \frac {2 \, {\left (\frac {15 \, {\left (e x + d\right )}^{\frac {7}{2}} B c^{2} - 21 \, {\left (5 \, B c^{2} d - {\left (2 \, B b c + A c^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 35 \, {\left (10 \, B c^{2} d^{2} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 105 \, {\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )} \sqrt {e x + d}}{e^{5}} + \frac {35 \, {\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} - {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - 3 \, {\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{5}}\right )}}{105 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 316, normalized size = 1.20 \begin {gather*} \frac {{\left (d+e\,x\right )}^{5/2}\,\left (2\,A\,c^2\,e-10\,B\,c^2\,d+4\,B\,b\,c\,e\right )}{5\,e^6}-\frac {\left (d+e\,x\right )\,\left (6\,B\,b^2\,d^2\,e^2-4\,A\,b^2\,d\,e^3-16\,B\,b\,c\,d^3\,e+12\,A\,b\,c\,d^2\,e^2+10\,B\,c^2\,d^4-8\,A\,c^2\,d^3\,e\right )-\frac {2\,B\,c^2\,d^5}{3}+\frac {2\,A\,c^2\,d^4\,e}{3}+\frac {2\,A\,b^2\,d^2\,e^3}{3}-\frac {2\,B\,b^2\,d^3\,e^2}{3}+\frac {4\,B\,b\,c\,d^4\,e}{3}-\frac {4\,A\,b\,c\,d^3\,e^2}{3}}{e^6\,{\left (d+e\,x\right )}^{3/2}}+\frac {\sqrt {d+e\,x}\,\left (-6\,B\,b^2\,d\,e^2+2\,A\,b^2\,e^3+24\,B\,b\,c\,d^2\,e-12\,A\,b\,c\,d\,e^2-20\,B\,c^2\,d^3+12\,A\,c^2\,d^2\,e\right )}{e^6}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (2\,B\,b^2\,e^2-16\,B\,b\,c\,d\,e+4\,A\,b\,c\,e^2+20\,B\,c^2\,d^2-8\,A\,c^2\,d\,e\right )}{3\,e^6}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 75.51, size = 292, normalized size = 1.11 \begin {gather*} \frac {2 B c^{2} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{6}} + \frac {2 d^{2} \left (- A e + B d\right ) \left (b e - c d\right )^{2}}{3 e^{6} \left (d + e x\right )^{\frac {3}{2}}} - \frac {2 d \left (b e - c d\right ) \left (- 2 A b e^{2} + 4 A c d e + 3 B b d e - 5 B c d^{2}\right )}{e^{6} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (2 A c^{2} e + 4 B b c e - 10 B c^{2} d\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (4 A b c e^{2} - 8 A c^{2} d e + 2 B b^{2} e^{2} - 16 B b c d e + 20 B c^{2} d^{2}\right )}{3 e^{6}} + \frac {\sqrt {d + e x} \left (2 A b^{2} e^{3} - 12 A b c d e^{2} + 12 A c^{2} d^{2} e - 6 B b^{2} d e^{2} + 24 B b c d^{2} e - 20 B c^{2} d^{3}\right )}{e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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