Optimal. Leaf size=403 \[ \frac {f (g+h x)^{1+m} \left (a+c x^2\right )^{3/2}}{c h (4+m)}-\frac {\left (a f h^2 (1+m)-c \left (3 f g^2-h (e g-d h) (4+m)\right )\right ) (g+h x)^{1+m} \sqrt {a+c x^2} F_1\left (1+m;-\frac {1}{2},-\frac {1}{2};2+m;\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}},\frac {g+h x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}\right )}{c h^3 (1+m) (4+m) \sqrt {1-\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}}} \sqrt {1-\frac {g+h x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}}}-\frac {(3 f g-e h (4+m)) (g+h x)^{2+m} \sqrt {a+c x^2} F_1\left (2+m;-\frac {1}{2},-\frac {1}{2};3+m;\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}},\frac {g+h x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}\right )}{h^3 (2+m) (4+m) \sqrt {1-\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}}} \sqrt {1-\frac {g+h x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}}} \]
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Rubi [A]
time = 0.28, antiderivative size = 401, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1668, 858, 774,
138} \begin {gather*} \frac {\sqrt {a+c x^2} (g+h x)^{m+1} F_1\left (m+1;-\frac {1}{2},-\frac {1}{2};m+2;\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}},\frac {g+h x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}\right ) \left (-a f h^2 (m+1)-c h (m+4) (e g-d h)+3 c f g^2\right )}{c h^3 (m+1) (m+4) \sqrt {1-\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}}} \sqrt {1-\frac {g+h x}{\frac {\sqrt {-a} h}{\sqrt {c}}+g}}}-\frac {\sqrt {a+c x^2} (g+h x)^{m+2} (3 f g-e h (m+4)) F_1\left (m+2;-\frac {1}{2},-\frac {1}{2};m+3;\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}},\frac {g+h x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}\right )}{h^3 (m+2) (m+4) \sqrt {1-\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}}} \sqrt {1-\frac {g+h x}{\frac {\sqrt {-a} h}{\sqrt {c}}+g}}}+\frac {f \left (a+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 774
Rule 858
Rule 1668
Rubi steps
\begin {align*} \int (g+h x)^m \sqrt {a+c x^2} \left (d+e x+f x^2\right ) \, dx &=\frac {f (g+h x)^{1+m} \left (a+c x^2\right )^{3/2}}{c h (4+m)}+\frac {\int (g+h x)^m \left (-h^2 (a f (1+m)-c d (4+m))-c h (3 f g-e h (4+m)) x\right ) \sqrt {a+c x^2} \, dx}{c h^2 (4+m)}\\ &=\frac {f (g+h x)^{1+m} \left (a+c x^2\right )^{3/2}}{c h (4+m)}-\frac {(3 f g-e h (4+m)) \int (g+h x)^{1+m} \sqrt {a+c x^2} \, dx}{h^2 (4+m)}+\frac {\left (3 c f g^2-a f h^2 (1+m)-c h (e g-d h) (4+m)\right ) \int (g+h x)^m \sqrt {a+c x^2} \, dx}{c h^2 (4+m)}\\ &=\frac {f (g+h x)^{1+m} \left (a+c x^2\right )^{3/2}}{c h (4+m)}-\frac {\left ((3 f g-e h (4+m)) \sqrt {a+c x^2}\right ) \text {Subst}\left (\int x^{1+m} \sqrt {1-\frac {x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}}} \sqrt {1-\frac {x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}} \, dx,x,g+h x\right )}{h^3 (4+m) \sqrt {1-\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}}} \sqrt {1-\frac {g+h x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}}}+\frac {\left (\left (3 c f g^2-a f h^2 (1+m)-c h (e g-d h) (4+m)\right ) \sqrt {a+c x^2}\right ) \text {Subst}\left (\int x^m \sqrt {1-\frac {x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}}} \sqrt {1-\frac {x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}} \, dx,x,g+h x\right )}{c h^3 (4+m) \sqrt {1-\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}}} \sqrt {1-\frac {g+h x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}}}\\ &=\frac {f (g+h x)^{1+m} \left (a+c x^2\right )^{3/2}}{c h (4+m)}+\frac {\left (3 c f g^2-a f h^2 (1+m)-c h (e g-d h) (4+m)\right ) (g+h x)^{1+m} \sqrt {a+c x^2} F_1\left (1+m;-\frac {1}{2},-\frac {1}{2};2+m;\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}},\frac {g+h x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}\right )}{c h^3 (1+m) (4+m) \sqrt {1-\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}}} \sqrt {1-\frac {g+h x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}}}-\frac {(3 f g-e h (4+m)) (g+h x)^{2+m} \sqrt {a+c x^2} F_1\left (2+m;-\frac {1}{2},-\frac {1}{2};3+m;\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}},\frac {g+h x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}\right )}{h^3 (2+m) (4+m) \sqrt {1-\frac {g+h x}{g-\frac {\sqrt {-a} h}{\sqrt {c}}}} \sqrt {1-\frac {g+h x}{g+\frac {\sqrt {-a} h}{\sqrt {c}}}}}\\ \end {align*}
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Mathematica [F]
time = 0.76, size = 0, normalized size = 0.00 \begin {gather*} \int (g+h x)^m \sqrt {a+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 = 0.02, size = 0, normalized size = 0.00 \[\int \left (h x +g \right )^{m} \left (f \,x^{2}+e x +d \right ) \sqrt {c \,x^{2}+a}\, dx\]
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 \sqrt {a + c x^{2}} \left (g + h x\right )^{m} \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+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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