Optimal. Leaf size=250 \[ \frac {\left (c d^2-a e^2\right ) (d+e x)^{n+1}}{c^2 e^3 (n+1)}+\frac {(-a)^{3/2} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 c^2 (n+1) \left (\sqrt {c} d-\sqrt {-a} e\right )}-\frac {(-a)^{3/2} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 c^2 (n+1) \left (\sqrt {-a} e+\sqrt {c} d\right )}-\frac {2 d (d+e x)^{n+2}}{c e^3 (n+2)}+\frac {(d+e x)^{n+3}}{c e^3 (n+3)} \]
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Rubi [A] time = 0.40, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1629, 712, 68} \[ \frac {\left (c d^2-a e^2\right ) (d+e x)^{n+1}}{c^2 e^3 (n+1)}+\frac {(-a)^{3/2} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 c^2 (n+1) \left (\sqrt {c} d-\sqrt {-a} e\right )}-\frac {(-a)^{3/2} (d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 c^2 (n+1) \left (\sqrt {-a} e+\sqrt {c} d\right )}-\frac {2 d (d+e x)^{n+2}}{c e^3 (n+2)}+\frac {(d+e x)^{n+3}}{c e^3 (n+3)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 712
Rule 1629
Rubi steps
\begin {align*} \int \frac {x^4 (d+e x)^n}{a+c x^2} \, dx &=\int \left (\frac {\left (c d^2-a e^2\right ) (d+e x)^n}{c^2 e^2}-\frac {2 d (d+e x)^{1+n}}{c e^2}+\frac {(d+e x)^{2+n}}{c e^2}+\frac {a^2 (d+e x)^n}{c^2 \left (a+c x^2\right )}\right ) \, dx\\ &=\frac {\left (c d^2-a e^2\right ) (d+e x)^{1+n}}{c^2 e^3 (1+n)}-\frac {2 d (d+e x)^{2+n}}{c e^3 (2+n)}+\frac {(d+e x)^{3+n}}{c e^3 (3+n)}+\frac {a^2 \int \frac {(d+e x)^n}{a+c x^2} \, dx}{c^2}\\ &=\frac {\left (c d^2-a e^2\right ) (d+e x)^{1+n}}{c^2 e^3 (1+n)}-\frac {2 d (d+e x)^{2+n}}{c e^3 (2+n)}+\frac {(d+e x)^{3+n}}{c e^3 (3+n)}+\frac {a^2 \int \left (\frac {\sqrt {-a} (d+e x)^n}{2 a \left (\sqrt {-a}-\sqrt {c} x\right )}+\frac {\sqrt {-a} (d+e x)^n}{2 a \left (\sqrt {-a}+\sqrt {c} x\right )}\right ) \, dx}{c^2}\\ &=\frac {\left (c d^2-a e^2\right ) (d+e x)^{1+n}}{c^2 e^3 (1+n)}-\frac {2 d (d+e x)^{2+n}}{c e^3 (2+n)}+\frac {(d+e x)^{3+n}}{c e^3 (3+n)}-\frac {(-a)^{3/2} \int \frac {(d+e x)^n}{\sqrt {-a}-\sqrt {c} x} \, dx}{2 c^2}-\frac {(-a)^{3/2} \int \frac {(d+e x)^n}{\sqrt {-a}+\sqrt {c} x} \, dx}{2 c^2}\\ &=\frac {\left (c d^2-a e^2\right ) (d+e x)^{1+n}}{c^2 e^3 (1+n)}-\frac {2 d (d+e x)^{2+n}}{c e^3 (2+n)}+\frac {(d+e x)^{3+n}}{c e^3 (3+n)}+\frac {(-a)^{3/2} (d+e x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 c^2 \left (\sqrt {c} d-\sqrt {-a} e\right ) (1+n)}-\frac {(-a)^{3/2} (d+e x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 c^2 \left (\sqrt {c} d+\sqrt {-a} e\right ) (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 217, normalized size = 0.87 \[ \frac {(d+e x)^{n+1} \left (\frac {2 \left (c d^2-a e^2\right )}{e^3 (n+1)}+\frac {(-a)^{3/2} \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{(n+1) \left (\sqrt {c} d-\sqrt {-a} e\right )}+\frac {\sqrt {-a} a \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{(n+1) \left (\sqrt {-a} e+\sqrt {c} d\right )}+\frac {2 c (d+e x)^2}{e^3 (n+3)}-\frac {4 c d (d+e x)}{e^3 (n+2)}\right )}{2 c^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )}^{n} x^{4}}{c x^{2} + a}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{n} x^{4}}{c x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \left (e x +d \right )^{n}}{c \,x^{2}+a}\, dx \]
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 \[ \int \frac {{\left (e x + d\right )}^{n} x^{4}}{c x^{2} + a}\,{d x} \]
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 \[ \int \frac {x^4\,{\left (d+e\,x\right )}^n}{c\,x^2+a} \,d x \]
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 \[ \int \frac {x^{4} \left (d + e x\right )^{n}}{a + c x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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