Optimal. Leaf size=173 \[ \frac {2 (f+g x)^{3/2} \left (a e^2 g^2+c \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )\right )}{3 g^5}-\frac {2 \left (a g^2+c f^2\right ) (e f-d g)^2}{g^5 \sqrt {f+g x}}-\frac {4 \sqrt {f+g x} (e f-d g) \left (a e g^2+c f (2 e f-d g)\right )}{g^5}-\frac {4 c e (f+g x)^{5/2} (2 e f-d g)}{5 g^5}+\frac {2 c e^2 (f+g x)^{7/2}}{7 g^5} \]
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Rubi [A] time = 0.20, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {898, 1261} \[ \frac {2 (f+g x)^{3/2} \left (a e^2 g^2+c \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )\right )}{3 g^5}-\frac {2 \left (a g^2+c f^2\right ) (e f-d g)^2}{g^5 \sqrt {f+g x}}-\frac {4 \sqrt {f+g x} (e f-d g) \left (a e g^2+c f (2 e f-d g)\right )}{g^5}-\frac {4 c e (f+g x)^{5/2} (2 e f-d g)}{5 g^5}+\frac {2 c e^2 (f+g x)^{7/2}}{7 g^5} \]
Antiderivative was successfully verified.
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Rule 898
Rule 1261
Rubi steps
\begin {align*} \int \frac {(d+e x)^2 \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {\left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )^2 \left (\frac {c f^2+a g^2}{g^2}-\frac {2 c f x^2}{g^2}+\frac {c x^4}{g^2}\right )}{x^2} \, dx,x,\sqrt {f+g x}\right )}{g}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (\frac {2 (e f-d g) \left (-a e g^2-c f (2 e f-d g)\right )}{g^4}+\frac {(-e f+d g)^2 \left (c f^2+a g^2\right )}{g^4 x^2}+\frac {\left (a e^2 g^2+c \left (6 e^2 f^2-6 d e f g+d^2 g^2\right )\right ) x^2}{g^4}-\frac {2 c e (2 e f-d g) x^4}{g^4}+\frac {c e^2 x^6}{g^4}\right ) \, dx,x,\sqrt {f+g x}\right )}{g}\\ &=-\frac {2 (e f-d g)^2 \left (c f^2+a g^2\right )}{g^5 \sqrt {f+g x}}-\frac {4 (e f-d g) \left (a e g^2+c f (2 e f-d g)\right ) \sqrt {f+g x}}{g^5}+\frac {2 \left (a e^2 g^2+c \left (6 e^2 f^2-6 d e f g+d^2 g^2\right )\right ) (f+g x)^{3/2}}{3 g^5}-\frac {4 c e (2 e f-d g) (f+g x)^{5/2}}{5 g^5}+\frac {2 c e^2 (f+g x)^{7/2}}{7 g^5}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 149, normalized size = 0.86 \[ \frac {2 \left (35 (f+g x)^2 \left (a e^2 g^2+c \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )\right )-105 \left (a g^2+c f^2\right ) (e f-d g)^2-210 (f+g x) (e f-d g) \left (a e g^2+c f (2 e f-d g)\right )-42 c e (f+g x)^3 (2 e f-d g)+15 c e^2 (f+g x)^4\right )}{105 g^5 \sqrt {f+g x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 206, normalized size = 1.19 \[ \frac {2 \, {\left (15 \, c e^{2} g^{4} x^{4} - 384 \, c e^{2} f^{4} + 672 \, c d e f^{3} g + 420 \, a d e f g^{3} - 105 \, a d^{2} g^{4} - 280 \, {\left (c d^{2} + a e^{2}\right )} f^{2} g^{2} - 6 \, {\left (4 \, c e^{2} f g^{3} - 7 \, c d e g^{4}\right )} x^{3} + {\left (48 \, c e^{2} f^{2} g^{2} - 84 \, c d e f g^{3} + 35 \, {\left (c d^{2} + a e^{2}\right )} g^{4}\right )} x^{2} - 2 \, {\left (96 \, c e^{2} f^{3} g - 168 \, c d e f^{2} g^{2} - 105 \, a d e g^{4} + 70 \, {\left (c d^{2} + a e^{2}\right )} f g^{3}\right )} x\right )} \sqrt {g x + f}}{105 \, {\left (g^{6} x + f g^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 275, normalized size = 1.59 \[ -\frac {2 \, {\left (c d^{2} f^{2} g^{2} + a d^{2} g^{4} - 2 \, c d f^{3} g e - 2 \, a d f g^{3} e + c f^{4} e^{2} + a f^{2} g^{2} e^{2}\right )}}{\sqrt {g x + f} g^{5}} + \frac {2 \, {\left (35 \, {\left (g x + f\right )}^{\frac {3}{2}} c d^{2} g^{32} - 210 \, \sqrt {g x + f} c d^{2} f g^{32} + 42 \, {\left (g x + f\right )}^{\frac {5}{2}} c d g^{31} e - 210 \, {\left (g x + f\right )}^{\frac {3}{2}} c d f g^{31} e + 630 \, \sqrt {g x + f} c d f^{2} g^{31} e + 210 \, \sqrt {g x + f} a d g^{33} e + 15 \, {\left (g x + f\right )}^{\frac {7}{2}} c g^{30} e^{2} - 84 \, {\left (g x + f\right )}^{\frac {5}{2}} c f g^{30} e^{2} + 210 \, {\left (g x + f\right )}^{\frac {3}{2}} c f^{2} g^{30} e^{2} - 420 \, \sqrt {g x + f} c f^{3} g^{30} e^{2} + 35 \, {\left (g x + f\right )}^{\frac {3}{2}} a g^{32} e^{2} - 210 \, \sqrt {g x + f} a f g^{32} e^{2}\right )}}{105 \, g^{35}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 215, normalized size = 1.24 \[ -\frac {2 \left (-15 e^{2} c \,x^{4} g^{4}-42 c d e \,g^{4} x^{3}+24 c \,e^{2} f \,g^{3} x^{3}-35 a \,e^{2} g^{4} x^{2}-35 c \,d^{2} g^{4} x^{2}+84 c d e f \,g^{3} x^{2}-48 c \,e^{2} f^{2} g^{2} x^{2}-210 a d e \,g^{4} x +140 a \,e^{2} f \,g^{3} x +140 c \,d^{2} f \,g^{3} x -336 c d e \,f^{2} g^{2} x +192 c \,e^{2} f^{3} g x +105 d^{2} a \,g^{4}-420 a d e f \,g^{3}+280 a \,e^{2} f^{2} g^{2}+280 c \,d^{2} f^{2} g^{2}-672 c d e \,f^{3} g +384 c \,e^{2} f^{4}\right )}{105 \sqrt {g x +f}\, g^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 205, normalized size = 1.18 \[ \frac {2 \, {\left (\frac {15 \, {\left (g x + f\right )}^{\frac {7}{2}} c e^{2} - 42 \, {\left (2 \, c e^{2} f - c d e g\right )} {\left (g x + f\right )}^{\frac {5}{2}} + 35 \, {\left (6 \, c e^{2} f^{2} - 6 \, c d e f g + {\left (c d^{2} + a e^{2}\right )} g^{2}\right )} {\left (g x + f\right )}^{\frac {3}{2}} - 210 \, {\left (2 \, c e^{2} f^{3} - 3 \, c d e f^{2} g - a d e g^{3} + {\left (c d^{2} + a e^{2}\right )} f g^{2}\right )} \sqrt {g x + f}}{g^{4}} - \frac {105 \, {\left (c e^{2} f^{4} - 2 \, c d e f^{3} g - 2 \, a d e f g^{3} + a d^{2} g^{4} + {\left (c d^{2} + a e^{2}\right )} f^{2} g^{2}\right )}}{\sqrt {g x + f} g^{4}}\right )}}{105 \, g} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.66, size = 199, normalized size = 1.15 \[ \frac {{\left (f+g\,x\right )}^{3/2}\,\left (2\,c\,d^2\,g^2-12\,c\,d\,e\,f\,g+12\,c\,e^2\,f^2+2\,a\,e^2\,g^2\right )}{3\,g^5}-\frac {2\,c\,d^2\,f^2\,g^2+2\,a\,d^2\,g^4-4\,c\,d\,e\,f^3\,g-4\,a\,d\,e\,f\,g^3+2\,c\,e^2\,f^4+2\,a\,e^2\,f^2\,g^2}{g^5\,\sqrt {f+g\,x}}+\frac {4\,\sqrt {f+g\,x}\,\left (d\,g-e\,f\right )\,\left (2\,c\,e\,f^2-c\,d\,f\,g+a\,e\,g^2\right )}{g^5}+\frac {2\,c\,e^2\,{\left (f+g\,x\right )}^{7/2}}{7\,g^5}+\frac {4\,c\,e\,{\left (f+g\,x\right )}^{5/2}\,\left (d\,g-2\,e\,f\right )}{5\,g^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 50.85, size = 204, normalized size = 1.18 \[ \frac {2 c e^{2} \left (f + g x\right )^{\frac {7}{2}}}{7 g^{5}} + \frac {\left (f + g x\right )^{\frac {5}{2}} \left (4 c d e g - 8 c e^{2} f\right )}{5 g^{5}} + \frac {\left (f + g x\right )^{\frac {3}{2}} \left (2 a e^{2} g^{2} + 2 c d^{2} g^{2} - 12 c d e f g + 12 c e^{2} f^{2}\right )}{3 g^{5}} + \frac {\sqrt {f + g x} \left (4 a d e g^{3} - 4 a e^{2} f g^{2} - 4 c d^{2} f g^{2} + 12 c d e f^{2} g - 8 c e^{2} f^{3}\right )}{g^{5}} - \frac {2 \left (a g^{2} + c f^{2}\right ) \left (d g - e f\right )^{2}}{g^{5} \sqrt {f + g x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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