Optimal. Leaf size=313 \[ \frac {\left (\frac {b \left (b^2-3 a c\right )}{c \sqrt {b^2-4 a c}}+a-\frac {b^2}{c}\right ) \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac {2 c \left (e x^2+d\right )}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )}{2 c (q+1) \left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right )}+\frac {\left (-\frac {b \left (b^2-3 a c\right )}{c \sqrt {b^2-4 a c}}+a-\frac {b^2}{c}\right ) \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac {2 c \left (e x^2+d\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{2 c (q+1) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {(b e+c d) \left (d+e x^2\right )^{q+1}}{2 c^2 e^2 (q+1)}+\frac {\left (d+e x^2\right )^{q+2}}{2 c e^2 (q+2)} \]
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Rubi [A] time = 0.94, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1251, 1628, 68} \[ \frac {\left (\frac {b \left (b^2-3 a c\right )}{c \sqrt {b^2-4 a c}}+a-\frac {b^2}{c}\right ) \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac {2 c \left (e x^2+d\right )}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )}{2 c (q+1) \left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right )}+\frac {\left (-\frac {b \left (b^2-3 a c\right )}{c \sqrt {b^2-4 a c}}+a-\frac {b^2}{c}\right ) \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac {2 c \left (e x^2+d\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{2 c (q+1) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {(b e+c d) \left (d+e x^2\right )^{q+1}}{2 c^2 e^2 (q+1)}+\frac {\left (d+e x^2\right )^{q+2}}{2 c e^2 (q+2)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 1251
Rule 1628
Rubi steps
\begin {align*} \int \frac {x^7 \left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3 (d+e x)^q}{a+b x+c x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {(-c d-b e) (d+e x)^q}{c^2 e}+\frac {\left (\frac {b^2}{c^2}-\frac {a}{c}-\frac {b \left (b^2-3 a c\right )}{c^2 \sqrt {b^2-4 a c}}\right ) (d+e x)^q}{b-\sqrt {b^2-4 a c}+2 c x}+\frac {\left (\frac {b^2}{c^2}-\frac {a}{c}+\frac {b \left (b^2-3 a c\right )}{c^2 \sqrt {b^2-4 a c}}\right ) (d+e x)^q}{b+\sqrt {b^2-4 a c}+2 c x}+\frac {(d+e x)^{1+q}}{c e}\right ) \, dx,x,x^2\right )\\ &=-\frac {(c d+b e) \left (d+e x^2\right )^{1+q}}{2 c^2 e^2 (1+q)}+\frac {\left (d+e x^2\right )^{2+q}}{2 c e^2 (2+q)}-\frac {\left (a-\frac {b^2}{c}-\frac {b \left (b^2-3 a c\right )}{c \sqrt {b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {(d+e x)^q}{b+\sqrt {b^2-4 a c}+2 c x} \, dx,x,x^2\right )}{2 c}-\frac {\left (a-\frac {b^2}{c}+\frac {b \left (b^2-3 a c\right )}{c \sqrt {b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {(d+e x)^q}{b-\sqrt {b^2-4 a c}+2 c x} \, dx,x,x^2\right )}{2 c}\\ &=-\frac {(c d+b e) \left (d+e x^2\right )^{1+q}}{2 c^2 e^2 (1+q)}+\frac {\left (d+e x^2\right )^{2+q}}{2 c e^2 (2+q)}+\frac {\left (a-\frac {b^2}{c}+\frac {b \left (b^2-3 a c\right )}{c \sqrt {b^2-4 a c}}\right ) \left (d+e x^2\right )^{1+q} \, _2F_1\left (1,1+q;2+q;\frac {2 c \left (d+e x^2\right )}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )}{2 c \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) (1+q)}+\frac {\left (a-\frac {b^2}{c}-\frac {b \left (b^2-3 a c\right )}{c \sqrt {b^2-4 a c}}\right ) \left (d+e x^2\right )^{1+q} \, _2F_1\left (1,1+q;2+q;\frac {2 c \left (d+e x^2\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{2 c \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (1+q)}\\ \end {align*}
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Mathematica [A] time = 0.79, size = 272, normalized size = 0.87 \[ \frac {\left (d+e x^2\right )^{q+1} \left (\frac {c \left (\frac {b \left (b^2-3 a c\right )}{c \sqrt {b^2-4 a c}}+a-\frac {b^2}{c}\right ) \, _2F_1\left (1,q+1;q+2;\frac {2 c \left (e x^2+d\right )}{2 c d+\left (\sqrt {b^2-4 a c}-b\right ) e}\right )}{(q+1) \left (e \left (\sqrt {b^2-4 a c}-b\right )+2 c d\right )}+\frac {c \left (-\frac {b \left (b^2-3 a c\right )}{c \sqrt {b^2-4 a c}}+a-\frac {b^2}{c}\right ) \, _2F_1\left (1,q+1;q+2;\frac {2 c \left (e x^2+d\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{(q+1) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {b e+c d}{e^2 (q+1)}+\frac {c \left (d+e x^2\right )}{e^2 (q+2)}\right )}{2 c^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x^{2} + d\right )}^{q} x^{7}}{c x^{4} + b x^{2} + a}, x\right ) \]
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 \[ \int \frac {{\left (e x^{2} + d\right )}^{q} x^{7}}{c x^{4} + b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {x^{7} \left (e \,x^{2}+d \right )^{q}}{c \,x^{4}+b \,x^{2}+a}\, dx \]
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 \[ \int \frac {{\left (e x^{2} + d\right )}^{q} x^{7}}{c x^{4} + b x^{2} + a}\,{d x} \]
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 \[ \int \frac {x^7\,{\left (e\,x^2+d\right )}^q}{c\,x^4+b\,x^2+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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