Optimal. Leaf size=496 \[ \frac {f (g+h x)^{1+m} \left (a+b x+c x^2\right )^{3/2}}{c h (4+m)}+\frac {\left (f h (b g-a h) (1+m)+c \left (3 f g^2-h (e g-d h) (4+m)\right )\right ) (g+h x)^{1+m} \sqrt {a+b x+c x^2} F_1\left (1+m;-\frac {1}{2},-\frac {1}{2};2+m;\frac {2 c (g+h x)}{2 c g-\left (b-\sqrt {b^2-4 a c}\right ) h},\frac {2 c (g+h x)}{2 c g-\left (b+\sqrt {b^2-4 a c}\right ) h}\right )}{c h^3 (1+m) (4+m) \sqrt {1-\frac {2 c (g+h x)}{2 c g-\left (b-\sqrt {b^2-4 a c}\right ) h}} \sqrt {1-\frac {2 c (g+h x)}{2 c g-\left (b+\sqrt {b^2-4 a c}\right ) h}}}-\frac {(b f h (5+2 m)+c (6 f g-2 e h (4+m))) (g+h x)^{2+m} \sqrt {a+b x+c x^2} F_1\left (2+m;-\frac {1}{2},-\frac {1}{2};3+m;\frac {2 c (g+h x)}{2 c g-\left (b-\sqrt {b^2-4 a c}\right ) h},\frac {2 c (g+h x)}{2 c g-\left (b+\sqrt {b^2-4 a c}\right ) h}\right )}{2 c h^3 (2+m) (4+m) \sqrt {1-\frac {2 c (g+h x)}{2 c g-\left (b-\sqrt {b^2-4 a c}\right ) h}} \sqrt {1-\frac {2 c (g+h x)}{2 c g-\left (b+\sqrt {b^2-4 a c}\right ) h}}} \]
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Rubi [A]
time = 0.39, antiderivative size = 494, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1667, 857, 773,
138} \begin {gather*} \frac {\sqrt {a+b x+c x^2} (g+h x)^{m+1} F_1\left (m+1;-\frac {1}{2},-\frac {1}{2};m+2;\frac {2 c (g+h x)}{2 c g-\left (b-\sqrt {b^2-4 a c}\right ) h},\frac {2 c (g+h x)}{2 c g-\left (b+\sqrt {b^2-4 a c}\right ) h}\right ) \left (f h (m+1) (b g-a h)-c h (m+4) (e g-d h)+3 c f g^2\right )}{c h^3 (m+1) (m+4) \sqrt {1-\frac {2 c (g+h x)}{2 c g-h \left (b-\sqrt {b^2-4 a c}\right )}} \sqrt {1-\frac {2 c (g+h x)}{2 c g-h \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {\sqrt {a+b x+c x^2} (g+h x)^{m+2} (b f h (2 m+5)-2 c e h (m+4)+6 c f g) F_1\left (m+2;-\frac {1}{2},-\frac {1}{2};m+3;\frac {2 c (g+h x)}{2 c g-\left (b-\sqrt {b^2-4 a c}\right ) h},\frac {2 c (g+h x)}{2 c g-\left (b+\sqrt {b^2-4 a c}\right ) h}\right )}{2 c h^3 (m+2) (m+4) \sqrt {1-\frac {2 c (g+h x)}{2 c g-h \left (b-\sqrt {b^2-4 a c}\right )}} \sqrt {1-\frac {2 c (g+h x)}{2 c g-h \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {f \left (a+b x+c x^2\right )^{3/2} (g+h x)^{m+1}}{c h (m+4)} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 773
Rule 857
Rule 1667
Rubi steps
\begin {align*} \int (g+h x)^m \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx &=\frac {f (g+h x)^{1+m} \left (a+b x+c x^2\right )^{3/2}}{c h (4+m)}+\frac {\int (g+h x)^m \left (-\frac {1}{2} h (3 b f g+2 a f h (1+m)-2 c d h (4+m))-\frac {1}{2} h (6 c f g-2 c e h (4+m)+b f h (5+2 m)) x\right ) \sqrt {a+b x+c x^2} \, dx}{c h^2 (4+m)}\\ &=\frac {f (g+h x)^{1+m} \left (a+b x+c x^2\right )^{3/2}}{c h (4+m)}+\frac {\left (3 c f g^2+f h (b g-a h) (1+m)-c h (e g-d h) (4+m)\right ) \int (g+h x)^m \sqrt {a+b x+c x^2} \, dx}{c h^2 (4+m)}-\frac {(6 c f g-2 c e h (4+m)+b f h (5+2 m)) \int (g+h x)^{1+m} \sqrt {a+b x+c x^2} \, dx}{2 c h^2 (4+m)}\\ &=\frac {f (g+h x)^{1+m} \left (a+b x+c x^2\right )^{3/2}}{c h (4+m)}+\frac {\left (\left (3 c f g^2+f h (b g-a h) (1+m)-c h (e g-d h) (4+m)\right ) \sqrt {a+b x+c x^2}\right ) \text {Subst}\left (\int x^m \sqrt {1-\frac {2 c x}{2 c g-\left (b-\sqrt {b^2-4 a c}\right ) h}} \sqrt {1-\frac {2 c x}{2 c g-\left (b+\sqrt {b^2-4 a c}\right ) h}} \, dx,x,g+h x\right )}{c h^3 (4+m) \sqrt {1-\frac {g+h x}{g-\frac {\left (b-\sqrt {b^2-4 a c}\right ) h}{2 c}}} \sqrt {1-\frac {g+h x}{g-\frac {\left (b+\sqrt {b^2-4 a c}\right ) h}{2 c}}}}-\frac {\left ((6 c f g-2 c e h (4+m)+b f h (5+2 m)) \sqrt {a+b x+c x^2}\right ) \text {Subst}\left (\int x^{1+m} \sqrt {1-\frac {2 c x}{2 c g-\left (b-\sqrt {b^2-4 a c}\right ) h}} \sqrt {1-\frac {2 c x}{2 c g-\left (b+\sqrt {b^2-4 a c}\right ) h}} \, dx,x,g+h x\right )}{2 c h^3 (4+m) \sqrt {1-\frac {g+h x}{g-\frac {\left (b-\sqrt {b^2-4 a c}\right ) h}{2 c}}} \sqrt {1-\frac {g+h x}{g-\frac {\left (b+\sqrt {b^2-4 a c}\right ) h}{2 c}}}}\\ &=\frac {f (g+h x)^{1+m} \left (a+b x+c x^2\right )^{3/2}}{c h (4+m)}+\frac {\left (3 c f g^2+f h (b g-a h) (1+m)-c h (e g-d h) (4+m)\right ) (g+h x)^{1+m} \sqrt {a+b x+c x^2} F_1\left (1+m;-\frac {1}{2},-\frac {1}{2};2+m;\frac {2 c (g+h x)}{2 c g-\left (b-\sqrt {b^2-4 a c}\right ) h},\frac {2 c (g+h x)}{2 c g-\left (b+\sqrt {b^2-4 a c}\right ) h}\right )}{c h^3 (1+m) (4+m) \sqrt {1-\frac {2 c (g+h x)}{2 c g-\left (b-\sqrt {b^2-4 a c}\right ) h}} \sqrt {1-\frac {2 c (g+h x)}{2 c g-\left (b+\sqrt {b^2-4 a c}\right ) h}}}-\frac {(6 c f g-2 c e h (4+m)+b f h (5+2 m)) (g+h x)^{2+m} \sqrt {a+b x+c x^2} F_1\left (2+m;-\frac {1}{2},-\frac {1}{2};3+m;\frac {2 c (g+h x)}{2 c g-\left (b-\sqrt {b^2-4 a c}\right ) h},\frac {2 c (g+h x)}{2 c g-\left (b+\sqrt {b^2-4 a c}\right ) h}\right )}{2 c h^3 (2+m) (4+m) \sqrt {1-\frac {2 c (g+h x)}{2 c g-\left (b-\sqrt {b^2-4 a c}\right ) h}} \sqrt {1-\frac {2 c (g+h x)}{2 c g-\left (b+\sqrt {b^2-4 a c}\right ) h}}}\\ \end {align*}
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Mathematica [F]
time = 1.45, size = 0, normalized size = 0.00 \begin {gather*} \int (g+h x)^m \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \left (h x +g \right )^{m} \left (f \,x^{2}+e x +d \right ) \sqrt {c \,x^{2}+b x +a}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (g + h x\right )^{m} \sqrt {a + b x + c x^{2}} \left (d + e x + f x^{2}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (g+h\,x\right )}^m\,\sqrt {c\,x^2+b\,x+a}\,\left (f\,x^2+e\,x+d\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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