Optimal. Leaf size=349 \[ -\frac {(c+d x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^{3/4} \left (\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d\right ) (1+n)}-\frac {(c+d x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b^{3/4} \left (\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d\right ) (1+n)}-\frac {(c+d x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{3/4} \left (\sqrt [4]{b} c-\sqrt [4]{-a} d\right ) (1+n)}-\frac {(c+d x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{3/4} \left (\sqrt [4]{b} c+\sqrt [4]{-a} d\right ) (1+n)} \]
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Rubi [A]
time = 0.57, antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6857, 845, 70}
\begin {gather*} -\frac {(c+d x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^{3/4} (n+1) \left (\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d\right )}-\frac {(c+d x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b^{3/4} (n+1) \left (\sqrt {-\sqrt {-a}} d+\sqrt [4]{b} c\right )}-\frac {(c+d x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{3/4} (n+1) \left (\sqrt [4]{b} c-\sqrt [4]{-a} d\right )}-\frac {(c+d x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{3/4} (n+1) \left (\sqrt [4]{-a} d+\sqrt [4]{b} c\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 845
Rule 6857
Rubi steps
\begin {align*} \int \frac {x^3 (c+d x)^n}{a+b x^4} \, dx &=\int \left (\frac {x (c+d x)^n}{2 \left (-\sqrt {-a} \sqrt {b}+b x^2\right )}+\frac {x (c+d x)^n}{2 \left (\sqrt {-a} \sqrt {b}+b x^2\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {x (c+d x)^n}{-\sqrt {-a} \sqrt {b}+b x^2} \, dx+\frac {1}{2} \int \frac {x (c+d x)^n}{\sqrt {-a} \sqrt {b}+b x^2} \, dx\\ &=\frac {1}{2} \int \left (-\frac {(c+d x)^n}{2 b^{3/4} \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}+\frac {(c+d x)^n}{2 b^{3/4} \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}\right ) \, dx+\frac {1}{2} \int \left (-\frac {(c+d x)^n}{2 b^{3/4} \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}+\frac {(c+d x)^n}{2 b^{3/4} \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}\right ) \, dx\\ &=-\frac {\int \frac {(c+d x)^n}{\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x} \, dx}{4 b^{3/4}}-\frac {\int \frac {(c+d x)^n}{\sqrt [4]{-a}-\sqrt [4]{b} x} \, dx}{4 b^{3/4}}+\frac {\int \frac {(c+d x)^n}{\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x} \, dx}{4 b^{3/4}}+\frac {\int \frac {(c+d x)^n}{\sqrt [4]{-a}+\sqrt [4]{b} x} \, dx}{4 b^{3/4}}\\ &=-\frac {(c+d x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^{3/4} \left (\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d\right ) (1+n)}-\frac {(c+d x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b^{3/4} \left (\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d\right ) (1+n)}-\frac {(c+d x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{3/4} \left (\sqrt [4]{b} c-\sqrt [4]{-a} d\right ) (1+n)}-\frac {(c+d x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{3/4} \left (\sqrt [4]{b} c+\sqrt [4]{-a} d\right ) (1+n)}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.35, size = 274, normalized size = 0.79 \begin {gather*} \frac {(c+d x)^{1+n} \left (-\frac {\, _2F_1\left (1,1+n;2+n;\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}-\frac {\, _2F_1\left (1,1+n;2+n;\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}-\frac {\, _2F_1\left (1,1+n;2+n;\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}-\frac {\, _2F_1\left (1,1+n;2+n;\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{3/4} (1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \left (d x +c \right )^{n}}{b \,x^{4}+a}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 \frac {x^3\,{\left (c+d\,x\right )}^n}{b\,x^4+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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