Optimal. Leaf size=263 \[ -\frac {2 \sqrt {d+e x} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{e^6}-\frac {2 \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^6 \sqrt {d+e x}}+\frac {2 d^2 (B d-A e) (c d-b e)^2}{5 e^6 (d+e x)^{5/2}}-\frac {2 c (d+e x)^{3/2} (-A c e-2 b B e+5 B c d)}{3 e^6}-\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6 (d+e x)^{3/2}}+\frac {2 B c^2 (d+e x)^{5/2}}{5 e^6} \]
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Rubi [A] time = 0.16, 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 \sqrt {d+e x} \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 )}{e^6}-\frac {2 \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 \sqrt {d+e x}}+\frac {2 d^2 (B d-A e) (c d-b e)^2}{5 e^6 (d+e x)^{5/2}}-\frac {2 c (d+e x)^{3/2} (-A c e-2 b B e+5 B c d)}{3 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))}{3 e^6 (d+e x)^{3/2}}+\frac {2 B c^2 (d+e x)^{5/2}}{5 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)^{7/2}} \, dx &=\int \left (-\frac {d^2 (B d-A e) (c d-b e)^2}{e^5 (d+e x)^{7/2}}+\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)^{5/2}}+\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 (d+e x)^{3/2}}+\frac {-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 )}{e^5 \sqrt {d+e x}}+\frac {c (-5 B c d+2 b B e+A c e) \sqrt {d+e x}}{e^5}+\frac {B c^2 (d+e x)^{3/2}}{e^5}\right ) \, dx\\ &=\frac {2 d^2 (B d-A e) (c d-b e)^2}{5 e^6 (d+e x)^{5/2}}-\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6 (d+e x)^{3/2}}-\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 )}{e^6 \sqrt {d+e x}}-\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 ) \sqrt {d+e x}}{e^6}-\frac {2 c (5 B c d-2 b B e-A c e) (d+e x)^{3/2}}{3 e^6}+\frac {2 B c^2 (d+e x)^{5/2}}{5 e^6}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 272, normalized size = 1.03 \begin {gather*} \frac {2 \left (A e \left (-b^2 e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )+6 b c e \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )-\left (c^2 \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )\right )\right )+B \left (3 b^2 e^2 \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )-2 b c e \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )+c^2 \left (256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right )\right )\right )}{15 e^6 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.20, size = 399, normalized size = 1.52 \begin {gather*} \frac {2 \left (-3 A b^2 d^2 e^3+10 A b^2 d e^3 (d+e x)-15 A b^2 e^3 (d+e x)^2+6 A b c d^3 e^2-30 A b c d^2 e^2 (d+e x)+90 A b c d e^2 (d+e x)^2+30 A b c e^2 (d+e x)^3-3 A c^2 d^4 e+20 A c^2 d^3 e (d+e x)-90 A c^2 d^2 e (d+e x)^2-60 A c^2 d e (d+e x)^3+5 A c^2 e (d+e x)^4+3 b^2 B d^3 e^2-15 b^2 B d^2 e^2 (d+e x)+45 b^2 B d e^2 (d+e x)^2+15 b^2 B e^2 (d+e x)^3-6 b B c d^4 e+40 b B c d^3 e (d+e x)-180 b B c d^2 e (d+e x)^2-120 b B c d e (d+e x)^3+10 b B c e (d+e x)^4+3 B c^2 d^5-25 B c^2 d^4 (d+e x)+150 B c^2 d^3 (d+e x)^2+150 B c^2 d^2 (d+e x)^3-25 B c^2 d (d+e x)^4+3 B c^2 (d+e x)^5\right )}{15 e^6 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 322, normalized size = 1.22 \begin {gather*} \frac {2 \, {\left (3 \, B c^{2} e^{5} x^{5} + 256 \, B c^{2} d^{5} - 8 \, A b^{2} d^{2} e^{3} - 128 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 48 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - 5 \, {\left (2 \, B c^{2} d e^{4} - {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 5 \, {\left (16 \, B c^{2} d^{2} e^{3} - 8 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} + 15 \, {\left (32 \, B c^{2} d^{3} e^{2} - A b^{2} e^{5} - 16 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 6 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 20 \, {\left (32 \, B c^{2} d^{4} e - A b^{2} d e^{4} - 16 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 6 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 427, normalized size = 1.62 \begin {gather*} \frac {2}{15} \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} B c^{2} e^{24} - 25 \, {\left (x e + d\right )}^{\frac {3}{2}} B c^{2} d e^{24} + 150 \, \sqrt {x e + d} B c^{2} d^{2} e^{24} + 10 \, {\left (x e + d\right )}^{\frac {3}{2}} B b c e^{25} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{2} e^{25} - 120 \, \sqrt {x e + d} B b c d e^{25} - 60 \, \sqrt {x e + d} A c^{2} d e^{25} + 15 \, \sqrt {x e + d} B b^{2} e^{26} + 30 \, \sqrt {x e + d} A b c e^{26}\right )} e^{\left (-30\right )} + \frac {2 \, {\left (150 \, {\left (x e + d\right )}^{2} B c^{2} d^{3} - 25 \, {\left (x e + d\right )} B c^{2} d^{4} + 3 \, B c^{2} d^{5} - 180 \, {\left (x e + d\right )}^{2} B b c d^{2} e - 90 \, {\left (x e + d\right )}^{2} A c^{2} d^{2} e + 40 \, {\left (x e + d\right )} B b c d^{3} e + 20 \, {\left (x e + d\right )} A c^{2} d^{3} e - 6 \, B b c d^{4} e - 3 \, A c^{2} d^{4} e + 45 \, {\left (x e + d\right )}^{2} B b^{2} d e^{2} + 90 \, {\left (x e + d\right )}^{2} A b c d e^{2} - 15 \, {\left (x e + d\right )} B b^{2} d^{2} e^{2} - 30 \, {\left (x e + d\right )} A b c d^{2} e^{2} + 3 \, B b^{2} d^{3} e^{2} + 6 \, A b c d^{3} e^{2} - 15 \, {\left (x e + d\right )}^{2} A b^{2} e^{3} + 10 \, {\left (x e + d\right )} A b^{2} d e^{3} - 3 \, A b^{2} d^{2} e^{3}\right )} e^{\left (-6\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 = 0.06, size = 341, normalized size = 1.30 \begin {gather*} -\frac {2 \left (-3 B \,c^{2} x^{5} e^{5}-5 A \,c^{2} e^{5} x^{4}-10 B b c \,e^{5} x^{4}+10 B \,c^{2} d \,e^{4} x^{4}-30 A b c \,e^{5} x^{3}+40 A \,c^{2} d \,e^{4} x^{3}-15 B \,b^{2} e^{5} x^{3}+80 B b c d \,e^{4} x^{3}-80 B \,c^{2} d^{2} e^{3} x^{3}+15 A \,b^{2} e^{5} x^{2}-180 A b c d \,e^{4} x^{2}+240 A \,c^{2} d^{2} e^{3} x^{2}-90 B \,b^{2} d \,e^{4} x^{2}+480 B b c \,d^{2} e^{3} x^{2}-480 B \,c^{2} d^{3} e^{2} x^{2}+20 A \,b^{2} d \,e^{4} x -240 A b c \,d^{2} e^{3} x +320 A \,c^{2} d^{3} e^{2} x -120 B \,b^{2} d^{2} e^{3} x +640 B b c \,d^{3} e^{2} x -640 B \,c^{2} d^{4} e x +8 A \,b^{2} d^{2} e^{3}-96 A b c \,d^{3} e^{2}+128 A \,c^{2} d^{4} e -48 B \,b^{2} d^{3} e^{2}+256 B b c \,d^{4} e -256 B \,c^{2} d^{5}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 298, normalized size = 1.13 \begin {gather*} \frac {2 \, {\left (\frac {3 \, {\left (e x + d\right )}^{\frac {5}{2}} B c^{2} - 5 \, {\left (5 \, B c^{2} d - {\left (2 \, B b c + A c^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 15 \, {\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 )} \sqrt {e x + d}}{e^{5}} + \frac {3 \, B c^{2} d^{5} - 3 \, A b^{2} d^{2} e^{3} - 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + 15 \, {\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 )} {\left (e x + d\right )}^{2} - 5 \, {\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 )}}{{\left (e x + d\right )}^{\frac {5}{2}} e^{5}}\right )}}{15 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.56, size = 312, normalized size = 1.19 \begin {gather*} \frac {{\left (d+e\,x\right )}^{3/2}\,\left (2\,A\,c^2\,e-10\,B\,c^2\,d+4\,B\,b\,c\,e\right )}{3\,e^6}+\frac {\sqrt {d+e\,x}\,\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 )}{e^6}-\frac {\left (d+e\,x\right )\,\left (2\,B\,b^2\,d^2\,e^2-\frac {4\,A\,b^2\,d\,e^3}{3}-\frac {16\,B\,b\,c\,d^3\,e}{3}+4\,A\,b\,c\,d^2\,e^2+\frac {10\,B\,c^2\,d^4}{3}-\frac {8\,A\,c^2\,d^3\,e}{3}\right )+{\left (d+e\,x\right )}^2\,\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 )-\frac {2\,B\,c^2\,d^5}{5}+\frac {2\,A\,c^2\,d^4\,e}{5}+\frac {2\,A\,b^2\,d^2\,e^3}{5}-\frac {2\,B\,b^2\,d^3\,e^2}{5}+\frac {4\,B\,b\,c\,d^4\,e}{5}-\frac {4\,A\,b\,c\,d^3\,e^2}{5}}{e^6\,{\left (d+e\,x\right )}^{5/2}}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.86, size = 1833, normalized size = 6.97
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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