Optimal. Leaf size=212 \[ -\frac {2 (f+g x)^{5/2} \left (e g (-a e g-2 b d g+3 b e f)-c \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )\right )}{5 g^5}+\frac {2 \sqrt {f+g x} (e f-d g)^2 \left (a g^2-b f g+c f^2\right )}{g^5}-\frac {2 (f+g x)^{3/2} (e f-d g) (2 c f (2 e f-d g)-g (-2 a e g-b d g+3 b e f))}{3 g^5}-\frac {2 e (f+g x)^{7/2} (-b e g-2 c d g+4 c e f)}{7 g^5}+\frac {2 c e^2 (f+g x)^{9/2}}{9 g^5} \]
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Rubi [A] time = 0.34, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {897, 1153} \[ -\frac {2 (f+g x)^{5/2} \left (e g (-a e g-2 b d g+3 b e f)-c \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )\right )}{5 g^5}+\frac {2 \sqrt {f+g x} (e f-d g)^2 \left (a g^2-b f g+c f^2\right )}{g^5}-\frac {2 (f+g x)^{3/2} (e f-d g) (2 c f (2 e f-d g)-g (-2 a e g-b d g+3 b e f))}{3 g^5}-\frac {2 e (f+g x)^{7/2} (-b e g-2 c d g+4 c e f)}{7 g^5}+\frac {2 c e^2 (f+g x)^{9/2}}{9 g^5} \]
Antiderivative was successfully verified.
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Rule 897
Rule 1153
Rubi steps
\begin {align*} \int \frac {(d+e x)^2 \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )^2 \left (\frac {c f^2-b f g+a g^2}{g^2}-\frac {(2 c f-b g) x^2}{g^2}+\frac {c x^4}{g^2}\right ) \, dx,x,\sqrt {f+g x}\right )}{g}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (\frac {(-e f+d g)^2 \left (c f^2-b f g+a g^2\right )}{g^4}+\frac {(e f-d g) (-2 c f (2 e f-d g)+g (3 b e f-b d g-2 a e g)) x^2}{g^4}+\frac {\left (-e g (3 b e f-2 b d g-a e g)+c \left (6 e^2 f^2-6 d e f g+d^2 g^2\right )\right ) x^4}{g^4}+\frac {e (-4 c e f+2 c d g+b e g) x^6}{g^4}+\frac {c e^2 x^8}{g^4}\right ) \, dx,x,\sqrt {f+g x}\right )}{g}\\ &=\frac {2 (e f-d g)^2 \left (c f^2-b f g+a g^2\right ) \sqrt {f+g x}}{g^5}-\frac {2 (e f-d g) (2 c f (2 e f-d g)-g (3 b e f-b d g-2 a e g)) (f+g x)^{3/2}}{3 g^5}-\frac {2 \left (e g (3 b e f-2 b d g-a e g)-c \left (6 e^2 f^2-6 d e f g+d^2 g^2\right )\right ) (f+g x)^{5/2}}{5 g^5}-\frac {2 e (4 c e f-2 c d g-b e g) (f+g x)^{7/2}}{7 g^5}+\frac {2 c e^2 (f+g x)^{9/2}}{9 g^5}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 184, normalized size = 0.87 \[ \frac {2 \sqrt {f+g x} \left (-63 (f+g x)^2 \left (-e g (a e g+2 b d g-3 b e f)-c \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )\right )+315 (e f-d g)^2 \left (g (a g-b f)+c f^2\right )-105 (f+g x) (e f-d g) (g (2 a e g+b d g-3 b e f)+2 c f (2 e f-d g))-45 e (f+g x)^3 (-b e g-2 c d g+4 c e f)+35 c e^2 (f+g x)^4\right )}{315 g^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 260, normalized size = 1.23 \[ \frac {2 \, {\left (35 \, c e^{2} g^{4} x^{4} + 128 \, c e^{2} f^{4} + 315 \, a d^{2} g^{4} - 144 \, {\left (2 \, c d e + b e^{2}\right )} f^{3} g + 168 \, {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} f^{2} g^{2} - 210 \, {\left (b d^{2} + 2 \, a d e\right )} f g^{3} - 5 \, {\left (8 \, c e^{2} f g^{3} - 9 \, {\left (2 \, c d e + b e^{2}\right )} g^{4}\right )} x^{3} + 3 \, {\left (16 \, c e^{2} f^{2} g^{2} - 18 \, {\left (2 \, c d e + b e^{2}\right )} f g^{3} + 21 \, {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} g^{4}\right )} x^{2} - {\left (64 \, c e^{2} f^{3} g - 72 \, {\left (2 \, c d e + b e^{2}\right )} f^{2} g^{2} + 84 \, {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} f g^{3} - 105 \, {\left (b d^{2} + 2 \, a d e\right )} g^{4}\right )} x\right )} \sqrt {g x + f}}{315 \, g^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 363, normalized size = 1.71 \[ \frac {2 \, {\left (315 \, \sqrt {g x + f} a d^{2} + \frac {105 \, {\left ({\left (g x + f\right )}^{\frac {3}{2}} - 3 \, \sqrt {g x + f} f\right )} b d^{2}}{g} + \frac {210 \, {\left ({\left (g x + f\right )}^{\frac {3}{2}} - 3 \, \sqrt {g x + f} f\right )} a d e}{g} + \frac {21 \, {\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} f + 15 \, \sqrt {g x + f} f^{2}\right )} c d^{2}}{g^{2}} + \frac {42 \, {\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} f + 15 \, \sqrt {g x + f} f^{2}\right )} b d e}{g^{2}} + \frac {21 \, {\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} f + 15 \, \sqrt {g x + f} f^{2}\right )} a e^{2}}{g^{2}} + \frac {18 \, {\left (5 \, {\left (g x + f\right )}^{\frac {7}{2}} - 21 \, {\left (g x + f\right )}^{\frac {5}{2}} f + 35 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{2} - 35 \, \sqrt {g x + f} f^{3}\right )} c d e}{g^{3}} + \frac {9 \, {\left (5 \, {\left (g x + f\right )}^{\frac {7}{2}} - 21 \, {\left (g x + f\right )}^{\frac {5}{2}} f + 35 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{2} - 35 \, \sqrt {g x + f} f^{3}\right )} b e^{2}}{g^{3}} + \frac {{\left (35 \, {\left (g x + f\right )}^{\frac {9}{2}} - 180 \, {\left (g x + f\right )}^{\frac {7}{2}} f + 378 \, {\left (g x + f\right )}^{\frac {5}{2}} f^{2} - 420 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{3} + 315 \, \sqrt {g x + f} f^{4}\right )} c e^{2}}{g^{4}}\right )}}{315 \, g} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 315, normalized size = 1.49 \[ \frac {2 \sqrt {g x +f}\, \left (35 e^{2} c \,x^{4} g^{4}+45 b \,e^{2} g^{4} x^{3}+90 c d e \,g^{4} x^{3}-40 c \,e^{2} f \,g^{3} x^{3}+63 a \,e^{2} g^{4} x^{2}+126 b d e \,g^{4} x^{2}-54 b \,e^{2} f \,g^{3} x^{2}+63 c \,d^{2} g^{4} x^{2}-108 c d e f \,g^{3} x^{2}+48 c \,e^{2} f^{2} g^{2} x^{2}+210 a d e \,g^{4} x -84 a \,e^{2} f \,g^{3} x +105 b \,d^{2} g^{4} x -168 b d e f \,g^{3} x +72 b \,e^{2} f^{2} g^{2} x -84 c \,d^{2} f \,g^{3} x +144 c d e \,f^{2} g^{2} x -64 c \,e^{2} f^{3} g x +315 a \,d^{2} g^{4}-420 a d e f \,g^{3}+168 a \,e^{2} f^{2} g^{2}-210 b \,d^{2} f \,g^{3}+336 b d e \,f^{2} g^{2}-144 b \,e^{2} f^{3} g +168 c \,d^{2} f^{2} g^{2}-288 c d e \,f^{3} g +128 c \,e^{2} f^{4}\right )}{315 g^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 261, normalized size = 1.23 \[ \frac {2 \, {\left (35 \, {\left (g x + f\right )}^{\frac {9}{2}} c e^{2} - 45 \, {\left (4 \, c e^{2} f - {\left (2 \, c d e + b e^{2}\right )} g\right )} {\left (g x + f\right )}^{\frac {7}{2}} + 63 \, {\left (6 \, c e^{2} f^{2} - 3 \, {\left (2 \, c d e + b e^{2}\right )} f g + {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} g^{2}\right )} {\left (g x + f\right )}^{\frac {5}{2}} - 105 \, {\left (4 \, c e^{2} f^{3} - 3 \, {\left (2 \, c d e + b e^{2}\right )} f^{2} g + 2 \, {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} f g^{2} - {\left (b d^{2} + 2 \, a d e\right )} g^{3}\right )} {\left (g x + f\right )}^{\frac {3}{2}} + 315 \, {\left (c e^{2} f^{4} + a d^{2} g^{4} - {\left (2 \, c d e + b e^{2}\right )} f^{3} g + {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} f^{2} g^{2} - {\left (b d^{2} + 2 \, a d e\right )} f g^{3}\right )} \sqrt {g x + f}\right )}}{315 \, g^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.17, size = 204, normalized size = 0.96 \[ \frac {{\left (f+g\,x\right )}^{7/2}\,\left (2\,b\,e^2\,g-8\,c\,e^2\,f+4\,c\,d\,e\,g\right )}{7\,g^5}+\frac {{\left (f+g\,x\right )}^{5/2}\,\left (2\,c\,d^2\,g^2-12\,c\,d\,e\,f\,g+4\,b\,d\,e\,g^2+12\,c\,e^2\,f^2-6\,b\,e^2\,f\,g+2\,a\,e^2\,g^2\right )}{5\,g^5}+\frac {2\,{\left (f+g\,x\right )}^{3/2}\,\left (d\,g-e\,f\right )\,\left (2\,a\,e\,g^2+b\,d\,g^2+4\,c\,e\,f^2-3\,b\,e\,f\,g-2\,c\,d\,f\,g\right )}{3\,g^5}+\frac {2\,\sqrt {f+g\,x}\,{\left (d\,g-e\,f\right )}^2\,\left (c\,f^2-b\,f\,g+a\,g^2\right )}{g^5}+\frac {2\,c\,e^2\,{\left (f+g\,x\right )}^{9/2}}{9\,g^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 105.54, size = 1001, normalized size = 4.72 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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