Optimal. Leaf size=214 \[ -\frac{4 c \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 \sqrt{d+e x}}-\frac{2 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{5 e^6 (d+e x)^{5/2}}+\frac{2 \left (a e^2+c d^2\right )^2 (B d-A e)}{7 e^6 (d+e x)^{7/2}}+\frac{4 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{3 e^6 (d+e x)^{3/2}}-\frac{2 c^2 \sqrt{d+e x} (5 B d-A e)}{e^6}+\frac{2 B c^2 (d+e x)^{3/2}}{3 e^6} \]
[Out]
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Rubi [A] time = 0.2536, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ -\frac{4 c \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 \sqrt{d+e x}}-\frac{2 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{5 e^6 (d+e x)^{5/2}}+\frac{2 \left (a e^2+c d^2\right )^2 (B d-A e)}{7 e^6 (d+e x)^{7/2}}+\frac{4 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{3 e^6 (d+e x)^{3/2}}-\frac{2 c^2 \sqrt{d+e x} (5 B d-A e)}{e^6}+\frac{2 B c^2 (d+e x)^{3/2}}{3 e^6} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^(9/2),x]
[Out]
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Rubi in Sympy [A] time = 55.2132, size = 219, normalized size = 1.02 \[ \frac{2 B c^{2} \left (d + e x\right )^{\frac{3}{2}}}{3 e^{6}} + \frac{2 c^{2} \sqrt{d + e x} \left (A e - 5 B d\right )}{e^{6}} - \frac{4 c \left (- 2 A c d e + B a e^{2} + 5 B c d^{2}\right )}{e^{6} \sqrt{d + e x}} - \frac{4 c \left (A a e^{3} + 3 A c d^{2} e - 3 B a d e^{2} - 5 B c d^{3}\right )}{3 e^{6} \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (a e^{2} + c d^{2}\right ) \left (- 4 A c d e + B a e^{2} + 5 B c d^{2}\right )}{5 e^{6} \left (d + e x\right )^{\frac{5}{2}}} - \frac{2 \left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )^{2}}{7 e^{6} \left (d + e x\right )^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+a)**2/(e*x+d)**(9/2),x)
[Out]
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Mathematica [A] time = 0.607465, size = 180, normalized size = 0.84 \[ \frac{2 \sqrt{d+e x} \left (-\frac{210 c \left (a B e^2-2 A c d e+5 B c d^2\right )}{d+e x}-\frac{21 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{(d+e x)^3}+\frac{15 \left (a e^2+c d^2\right )^2 (B d-A e)}{(d+e x)^4}+\frac{70 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{(d+e x)^2}-35 c^2 (14 B d-3 A e)+35 B c^2 e x\right )}{105 e^6} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^(9/2),x]
[Out]
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Maple [A] time = 0.013, size = 259, normalized size = 1.2 \[ -{\frac{-70\,B{c}^{2}{x}^{5}{e}^{5}-210\,A{c}^{2}{e}^{5}{x}^{4}+700\,B{c}^{2}d{e}^{4}{x}^{4}-1680\,A{c}^{2}d{e}^{4}{x}^{3}+420\,Bac{e}^{5}{x}^{3}+5600\,B{c}^{2}{d}^{2}{e}^{3}{x}^{3}+140\,Aac{e}^{5}{x}^{2}-3360\,A{c}^{2}{d}^{2}{e}^{3}{x}^{2}+840\,Bacd{e}^{4}{x}^{2}+11200\,B{c}^{2}{d}^{3}{e}^{2}{x}^{2}+112\,Aacd{e}^{4}x-2688\,A{c}^{2}{d}^{3}{e}^{2}x+42\,B{a}^{2}{e}^{5}x+672\,Bac{d}^{2}{e}^{3}x+8960\,B{c}^{2}{d}^{4}ex+30\,A{a}^{2}{e}^{5}+32\,A{d}^{2}ac{e}^{3}-768\,A{d}^{4}{c}^{2}e+12\,Bd{a}^{2}{e}^{4}+192\,aBc{d}^{3}{e}^{2}+2560\,B{c}^{2}{d}^{5}}{105\,{e}^{6}} \left ( ex+d \right ) ^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+a)^2/(e*x+d)^(9/2),x)
[Out]
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Maxima [A] time = 0.69123, size = 344, normalized size = 1.61 \[ \frac{2 \,{\left (\frac{35 \,{\left ({\left (e x + d\right )}^{\frac{3}{2}} B c^{2} - 3 \,{\left (5 \, B c^{2} d - A c^{2} e\right )} \sqrt{e x + d}\right )}}{e^{5}} + \frac{15 \, B c^{2} d^{5} - 15 \, A c^{2} d^{4} e + 30 \, B a c d^{3} e^{2} - 30 \, A a c d^{2} e^{3} + 15 \, B a^{2} d e^{4} - 15 \, A a^{2} e^{5} - 210 \,{\left (5 \, B c^{2} d^{2} - 2 \, A c^{2} d e + B a c e^{2}\right )}{\left (e x + d\right )}^{3} + 70 \,{\left (5 \, B c^{2} d^{3} - 3 \, A c^{2} d^{2} e + 3 \, B a c d e^{2} - A a c e^{3}\right )}{\left (e x + d\right )}^{2} - 21 \,{\left (5 \, B c^{2} d^{4} - 4 \, A c^{2} d^{3} e + 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} + B a^{2} e^{4}\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{7}{2}} e^{5}}\right )}}{105 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)/(e*x + d)^(9/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.267047, size = 377, normalized size = 1.76 \[ \frac{2 \,{\left (35 \, B c^{2} e^{5} x^{5} - 1280 \, B c^{2} d^{5} + 384 \, A c^{2} d^{4} e - 96 \, B a c d^{3} e^{2} - 16 \, A a c d^{2} e^{3} - 6 \, B a^{2} d e^{4} - 15 \, A a^{2} e^{5} - 35 \,{\left (10 \, B c^{2} d e^{4} - 3 \, A c^{2} e^{5}\right )} x^{4} - 70 \,{\left (40 \, B c^{2} d^{2} e^{3} - 12 \, A c^{2} d e^{4} + 3 \, B a c e^{5}\right )} x^{3} - 70 \,{\left (80 \, B c^{2} d^{3} e^{2} - 24 \, A c^{2} d^{2} e^{3} + 6 \, B a c d e^{4} + A a c e^{5}\right )} x^{2} - 7 \,{\left (640 \, B c^{2} d^{4} e - 192 \, A c^{2} d^{3} e^{2} + 48 \, B a c d^{2} e^{3} + 8 \, A a c d e^{4} + 3 \, B a^{2} e^{5}\right )} x\right )}}{105 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )} \sqrt{e x + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)/(e*x + d)^(9/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 24.1571, size = 1855, normalized size = 8.67 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+a)**2/(e*x+d)**(9/2),x)
[Out]
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GIAC/XCAS [A] time = 0.310953, size = 427, normalized size = 2. \[ \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} B c^{2} e^{12} - 15 \, \sqrt{x e + d} B c^{2} d e^{12} + 3 \, \sqrt{x e + d} A c^{2} e^{13}\right )} e^{\left (-18\right )} - \frac{2 \,{\left (1050 \,{\left (x e + d\right )}^{3} B c^{2} d^{2} - 350 \,{\left (x e + d\right )}^{2} B c^{2} d^{3} + 105 \,{\left (x e + d\right )} B c^{2} d^{4} - 15 \, B c^{2} d^{5} - 420 \,{\left (x e + d\right )}^{3} A c^{2} d e + 210 \,{\left (x e + d\right )}^{2} A c^{2} d^{2} e - 84 \,{\left (x e + d\right )} A c^{2} d^{3} e + 15 \, A c^{2} d^{4} e + 210 \,{\left (x e + d\right )}^{3} B a c e^{2} - 210 \,{\left (x e + d\right )}^{2} B a c d e^{2} + 126 \,{\left (x e + d\right )} B a c d^{2} e^{2} - 30 \, B a c d^{3} e^{2} + 70 \,{\left (x e + d\right )}^{2} A a c e^{3} - 84 \,{\left (x e + d\right )} A a c d e^{3} + 30 \, A a c d^{2} e^{3} + 21 \,{\left (x e + d\right )} B a^{2} e^{4} - 15 \, B a^{2} d e^{4} + 15 \, A a^{2} e^{5}\right )} e^{\left (-6\right )}}{105 \,{\left (x e + d\right )}^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)/(e*x + d)^(9/2),x, algorithm="giac")
[Out]