Optimal. Leaf size=214 \[ -\frac {4 c \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 \sqrt {d+e x}}-\frac {2 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{5 e^6 (d+e x)^{5/2}}+\frac {2 \left (a e^2+c d^2\right )^2 (B d-A e)}{7 e^6 (d+e x)^{7/2}}+\frac {4 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{3 e^6 (d+e x)^{3/2}}-\frac {2 c^2 \sqrt {d+e x} (5 B d-A e)}{e^6}+\frac {2 B c^2 (d+e x)^{3/2}}{3 e^6} \]
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Rubi [A] time = 0.10, antiderivative size = 214, 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 = {772} \begin {gather*} -\frac {4 c \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 \sqrt {d+e x}}+\frac {4 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{3 e^6 (d+e x)^{3/2}}-\frac {2 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{5 e^6 (d+e x)^{5/2}}+\frac {2 \left (a e^2+c d^2\right )^2 (B d-A e)}{7 e^6 (d+e x)^{7/2}}-\frac {2 c^2 \sqrt {d+e x} (5 B d-A e)}{e^6}+\frac {2 B c^2 (d+e x)^{3/2}}{3 e^6} \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 )^2}{(d+e x)^{9/2}} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right )^2}{e^5 (d+e x)^{9/2}}+\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{e^5 (d+e x)^{7/2}}+\frac {2 c \left (-5 B c d^3+3 A c d^2 e-3 a B d e^2+a A e^3\right )}{e^5 (d+e x)^{5/2}}-\frac {2 c \left (-5 B c d^2+2 A c d e-a B e^2\right )}{e^5 (d+e x)^{3/2}}+\frac {c^2 (-5 B d+A e)}{e^5 \sqrt {d+e x}}+\frac {B c^2 \sqrt {d+e x}}{e^5}\right ) \, dx\\ &=\frac {2 (B d-A e) \left (c d^2+a e^2\right )^2}{7 e^6 (d+e x)^{7/2}}-\frac {2 \left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{5 e^6 (d+e x)^{5/2}}+\frac {4 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right )}{3 e^6 (d+e x)^{3/2}}-\frac {4 c \left (5 B c d^2-2 A c d e+a B e^2\right )}{e^6 \sqrt {d+e x}}-\frac {2 c^2 (5 B d-A e) \sqrt {d+e x}}{e^6}+\frac {2 B c^2 (d+e x)^{3/2}}{3 e^6}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 214, normalized size = 1.00 \begin {gather*} -\frac {2 \left (A e \left (15 a^2 e^4+2 a c e^2 \left (8 d^2+28 d e x+35 e^2 x^2\right )-3 c^2 \left (128 d^4+448 d^3 e x+560 d^2 e^2 x^2+280 d e^3 x^3+35 e^4 x^4\right )\right )+B \left (3 a^2 e^4 (2 d+7 e x)+6 a c e^2 \left (16 d^3+56 d^2 e x+70 d e^2 x^2+35 e^3 x^3\right )+5 c^2 \left (256 d^5+896 d^4 e x+1120 d^3 e^2 x^2+560 d^2 e^3 x^3+70 d e^4 x^4-7 e^5 x^5\right )\right )\right )}{105 e^6 (d+e x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 301, normalized size = 1.41 \begin {gather*} \frac {2 \left (-15 a^2 A e^5-21 a^2 B e^4 (d+e x)+15 a^2 B d e^4-30 a A c d^2 e^3+84 a A c d e^3 (d+e x)-70 a A c e^3 (d+e x)^2+30 a B c d^3 e^2-126 a B c d^2 e^2 (d+e x)+210 a B c d e^2 (d+e x)^2-210 a B c e^2 (d+e x)^3-15 A c^2 d^4 e+84 A c^2 d^3 e (d+e x)-210 A c^2 d^2 e (d+e x)^2+420 A c^2 d e (d+e x)^3+105 A c^2 e (d+e x)^4+15 B c^2 d^5-105 B c^2 d^4 (d+e x)+350 B c^2 d^3 (d+e x)^2-1050 B c^2 d^2 (d+e x)^3-525 B c^2 d (d+e x)^4+35 B c^2 (d+e x)^5\right )}{105 e^6 (d+e x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 290, normalized size = 1.36 \begin {gather*} \frac {2 \, {\left (35 \, B c^{2} e^{5} x^{5} - 1280 \, B c^{2} d^{5} + 384 \, A c^{2} d^{4} e - 96 \, B a c d^{3} e^{2} - 16 \, A a c d^{2} e^{3} - 6 \, B a^{2} d e^{4} - 15 \, A a^{2} e^{5} - 35 \, {\left (10 \, B c^{2} d e^{4} - 3 \, A c^{2} e^{5}\right )} x^{4} - 70 \, {\left (40 \, B c^{2} d^{2} e^{3} - 12 \, A c^{2} d e^{4} + 3 \, B a c e^{5}\right )} x^{3} - 70 \, {\left (80 \, B c^{2} d^{3} e^{2} - 24 \, A c^{2} d^{2} e^{3} + 6 \, B a c d e^{4} + A a c e^{5}\right )} x^{2} - 7 \, {\left (640 \, B c^{2} d^{4} e - 192 \, A c^{2} d^{3} e^{2} + 48 \, B a c d^{2} e^{3} + 8 \, A a c d e^{4} + 3 \, B a^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{105 \, {\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 316, normalized size = 1.48 \begin {gather*} \frac {2}{3} \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} B c^{2} e^{12} - 15 \, \sqrt {x e + d} B c^{2} d e^{12} + 3 \, \sqrt {x e + d} A c^{2} e^{13}\right )} e^{\left (-18\right )} - \frac {2 \, {\left (1050 \, {\left (x e + d\right )}^{3} B c^{2} d^{2} - 350 \, {\left (x e + d\right )}^{2} B c^{2} d^{3} + 105 \, {\left (x e + d\right )} B c^{2} d^{4} - 15 \, B c^{2} d^{5} - 420 \, {\left (x e + d\right )}^{3} A c^{2} d e + 210 \, {\left (x e + d\right )}^{2} A c^{2} d^{2} e - 84 \, {\left (x e + d\right )} A c^{2} d^{3} e + 15 \, A c^{2} d^{4} e + 210 \, {\left (x e + d\right )}^{3} B a c e^{2} - 210 \, {\left (x e + d\right )}^{2} B a c d e^{2} + 126 \, {\left (x e + d\right )} B a c d^{2} e^{2} - 30 \, B a c d^{3} e^{2} + 70 \, {\left (x e + d\right )}^{2} A a c e^{3} - 84 \, {\left (x e + d\right )} A a c d e^{3} + 30 \, A a c d^{2} e^{3} + 21 \, {\left (x e + d\right )} B a^{2} e^{4} - 15 \, B a^{2} d e^{4} + 15 \, A a^{2} e^{5}\right )} e^{\left (-6\right )}}{105 \, {\left (x e + d\right )}^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 259, normalized size = 1.21 \begin {gather*} -\frac {2 \left (-35 B \,c^{2} x^{5} e^{5}-105 A \,c^{2} e^{5} x^{4}+350 B \,c^{2} d \,e^{4} x^{4}-840 A \,c^{2} d \,e^{4} x^{3}+210 B a c \,e^{5} x^{3}+2800 B \,c^{2} d^{2} e^{3} x^{3}+70 A a c \,e^{5} x^{2}-1680 A \,c^{2} d^{2} e^{3} x^{2}+420 B a c d \,e^{4} x^{2}+5600 B \,c^{2} d^{3} e^{2} x^{2}+56 A a c d \,e^{4} x -1344 A \,c^{2} d^{3} e^{2} x +21 B \,a^{2} e^{5} x +336 B a c \,d^{2} e^{3} x +4480 B \,c^{2} d^{4} e x +15 A \,a^{2} e^{5}+16 A \,d^{2} a c \,e^{3}-384 A \,c^{2} d^{4} e +6 B \,a^{2} d \,e^{4}+96 B \,d^{3} a c \,e^{2}+1280 B \,c^{2} d^{5}\right )}{105 \left (e x +d \right )^{\frac {7}{2}} e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 255, normalized size = 1.19 \begin {gather*} \frac {2 \, {\left (\frac {35 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} B c^{2} - 3 \, {\left (5 \, B c^{2} d - A c^{2} e\right )} \sqrt {e x + d}\right )}}{e^{5}} + \frac {15 \, B c^{2} d^{5} - 15 \, A c^{2} d^{4} e + 30 \, B a c d^{3} e^{2} - 30 \, A a c d^{2} e^{3} + 15 \, B a^{2} d e^{4} - 15 \, A a^{2} e^{5} - 210 \, {\left (5 \, B c^{2} d^{2} - 2 \, A c^{2} d e + B a c e^{2}\right )} {\left (e x + d\right )}^{3} + 70 \, {\left (5 \, B c^{2} d^{3} - 3 \, A c^{2} d^{2} e + 3 \, B a c d e^{2} - A a c e^{3}\right )} {\left (e x + d\right )}^{2} - 21 \, {\left (5 \, B c^{2} d^{4} - 4 \, A c^{2} d^{3} e + 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} + B a^{2} e^{4}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {7}{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 = 1.82, size = 258, normalized size = 1.21 \begin {gather*} -\frac {2\,\left (6\,B\,a^2\,d\,e^4+21\,B\,a^2\,e^5\,x+15\,A\,a^2\,e^5+96\,B\,a\,c\,d^3\,e^2+336\,B\,a\,c\,d^2\,e^3\,x+16\,A\,a\,c\,d^2\,e^3+420\,B\,a\,c\,d\,e^4\,x^2+56\,A\,a\,c\,d\,e^4\,x+210\,B\,a\,c\,e^5\,x^3+70\,A\,a\,c\,e^5\,x^2+1280\,B\,c^2\,d^5+4480\,B\,c^2\,d^4\,e\,x-384\,A\,c^2\,d^4\,e+5600\,B\,c^2\,d^3\,e^2\,x^2-1344\,A\,c^2\,d^3\,e^2\,x+2800\,B\,c^2\,d^2\,e^3\,x^3-1680\,A\,c^2\,d^2\,e^3\,x^2+350\,B\,c^2\,d\,e^4\,x^4-840\,A\,c^2\,d\,e^4\,x^3-35\,B\,c^2\,e^5\,x^5-105\,A\,c^2\,e^5\,x^4\right )}{105\,e^6\,{\left (d+e\,x\right )}^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.15, size = 1855, normalized size = 8.67
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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