Optimal. Leaf size=466 \[ -\frac {2 d^2 \left (b \left (b^2 f-c (3 a f+c d)\right )-c x \left (2 a c f+b^2 (-f)+2 c^2 d\right )\right )}{f^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}+\frac {2 d (b+2 c x)}{f^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 b \sqrt {a+b x+c x^2}}{c f \left (b^2-4 a c\right )}-\frac {2 x (2 a+b x)}{f \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2} f}+\frac {d^{3/2} \tanh ^{-1}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 f \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2}}+\frac {d^{3/2} \tanh ^{-1}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 f \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2}} \]
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Rubi [A] time = 1.35, antiderivative size = 466, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {6725, 613, 738, 640, 621, 206, 975, 1033, 724} \begin {gather*} -\frac {2 d^2 \left (b \left (b^2 f-c (3 a f+c d)\right )-c x \left (2 a c f+b^2 (-f)+2 c^2 d\right )\right )}{f^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}+\frac {2 d (b+2 c x)}{f^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 b \sqrt {a+b x+c x^2}}{c f \left (b^2-4 a c\right )}-\frac {2 x (2 a+b x)}{f \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2} f}+\frac {d^{3/2} \tanh ^{-1}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 f \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2}}+\frac {d^{3/2} \tanh ^{-1}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 f \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 613
Rule 621
Rule 640
Rule 724
Rule 738
Rule 975
Rule 1033
Rule 6725
Rubi steps
\begin {align*} \int \frac {x^4}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx &=\int \left (-\frac {d}{f^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {x^2}{f \left (a+b x+c x^2\right )^{3/2}}+\frac {d^2}{f^2 \left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )}\right ) \, dx\\ &=-\frac {d \int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{f^2}+\frac {d^2 \int \frac {1}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx}{f^2}-\frac {\int \frac {x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{f}\\ &=-\frac {2 x (2 a+b x)}{\left (b^2-4 a c\right ) f \sqrt {a+b x+c x^2}}+\frac {2 d (b+2 c x)}{\left (b^2-4 a c\right ) f^2 \sqrt {a+b x+c x^2}}-\frac {2 d^2 \left (b \left (b^2 f-c (c d+3 a f)\right )-c \left (2 c^2 d-b^2 f+2 a c f\right ) x\right )}{\left (b^2-4 a c\right ) f^2 \left (b^2 d f-(c d+a f)^2\right ) \sqrt {a+b x+c x^2}}+\frac {2 \int \frac {2 a+b x}{\sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) f}-\frac {\left (2 d^2\right ) \int \frac {\frac {1}{2} \left (b^2-4 a c\right ) f (c d+a f)-\frac {1}{2} b \left (b^2-4 a c\right ) f^2 x}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx}{\left (b^2-4 a c\right ) f^2 \left (b^2 d f-(c d+a f)^2\right )}\\ &=-\frac {2 x (2 a+b x)}{\left (b^2-4 a c\right ) f \sqrt {a+b x+c x^2}}+\frac {2 d (b+2 c x)}{\left (b^2-4 a c\right ) f^2 \sqrt {a+b x+c x^2}}-\frac {2 d^2 \left (b \left (b^2 f-c (c d+3 a f)\right )-c \left (2 c^2 d-b^2 f+2 a c f\right ) x\right )}{\left (b^2-4 a c\right ) f^2 \left (b^2 d f-(c d+a f)^2\right ) \sqrt {a+b x+c x^2}}+\frac {2 b \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right ) f}-\frac {\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{c f}-\frac {d^{3/2} \int \frac {1}{\left (-\sqrt {d} \sqrt {f}-f x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 \sqrt {f} \left (c d-b \sqrt {d} \sqrt {f}+a f\right )}+\frac {d^{3/2} \int \frac {1}{\left (\sqrt {d} \sqrt {f}-f x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 \sqrt {f} \left (c d+b \sqrt {d} \sqrt {f}+a f\right )}\\ &=-\frac {2 x (2 a+b x)}{\left (b^2-4 a c\right ) f \sqrt {a+b x+c x^2}}+\frac {2 d (b+2 c x)}{\left (b^2-4 a c\right ) f^2 \sqrt {a+b x+c x^2}}-\frac {2 d^2 \left (b \left (b^2 f-c (c d+3 a f)\right )-c \left (2 c^2 d-b^2 f+2 a c f\right ) x\right )}{\left (b^2-4 a c\right ) f^2 \left (b^2 d f-(c d+a f)^2\right ) \sqrt {a+b x+c x^2}}+\frac {2 b \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right ) f}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{c f}+\frac {d^{3/2} \operatorname {Subst}\left (\int \frac {1}{4 c d f-4 b \sqrt {d} f^{3/2}+4 a f^2-x^2} \, dx,x,\frac {b \sqrt {d} \sqrt {f}-2 a f-\left (-2 c \sqrt {d} \sqrt {f}+b f\right ) x}{\sqrt {a+b x+c x^2}}\right )}{\sqrt {f} \left (c d-b \sqrt {d} \sqrt {f}+a f\right )}-\frac {d^{3/2} \operatorname {Subst}\left (\int \frac {1}{4 c d f+4 b \sqrt {d} f^{3/2}+4 a f^2-x^2} \, dx,x,\frac {-b \sqrt {d} \sqrt {f}-2 a f-\left (2 c \sqrt {d} \sqrt {f}+b f\right ) x}{\sqrt {a+b x+c x^2}}\right )}{\sqrt {f} \left (c d+b \sqrt {d} \sqrt {f}+a f\right )}\\ &=-\frac {2 x (2 a+b x)}{\left (b^2-4 a c\right ) f \sqrt {a+b x+c x^2}}+\frac {2 d (b+2 c x)}{\left (b^2-4 a c\right ) f^2 \sqrt {a+b x+c x^2}}-\frac {2 d^2 \left (b \left (b^2 f-c (c d+3 a f)\right )-c \left (2 c^2 d-b^2 f+2 a c f\right ) x\right )}{\left (b^2-4 a c\right ) f^2 \left (b^2 d f-(c d+a f)^2\right ) \sqrt {a+b x+c x^2}}+\frac {2 b \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right ) f}-\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2} f}+\frac {d^{3/2} \tanh ^{-1}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f \left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{3/2}}+\frac {d^{3/2} \tanh ^{-1}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f \left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 1.41, size = 562, normalized size = 1.21 \begin {gather*} \frac {\frac {f \left (a \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-2 \sqrt {c} \left (a (b-2 c x)+b c x^2\right ) \sqrt {a+x (b+c x)}\right )}{a c^{3/2} \left (4 a c-b^2\right )}+\frac {d^{3/2} f \left (\frac {\left (b^2-4 a c\right ) \left (a f+b \sqrt {d} \sqrt {f}+c d\right ) \tanh ^{-1}\left (\frac {-2 a \sqrt {f}+b \left (\sqrt {d}-\sqrt {f} x\right )+2 c \sqrt {d} x}{2 \sqrt {a+x (b+c x)} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{\sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}+\frac {\left (4 a c-b^2\right ) \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right ) \tanh ^{-1}\left (\frac {-2 \left (a \sqrt {f}+c \sqrt {d} x\right )-b \left (\sqrt {d}+\sqrt {f} x\right )}{2 \sqrt {a+x (b+c x)} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{\sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 \left (b^2-4 a c\right ) \left ((a f+c d)^2-b^2 d f\right )}+\frac {2 d (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)}}-\frac {2 f x^3 \left (-2 a c+b^2+b c x\right )}{a \left (b^2-4 a c\right ) \sqrt {a+x (b+c x)}}-\frac {2 d^2 \left (-b c (3 a f+c d)-2 c^2 x (a f+c d)+b^3 f+b^2 c f x\right )}{\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)} \left (b^2 d f-(a f+c d)^2\right )}}{f^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 1.54, size = 469, normalized size = 1.01 \begin {gather*} -\frac {d^2 \text {RootSum}\left [\text {$\#$1}^4 (-f)+2 \text {$\#$1}^2 a f+4 \text {$\#$1}^2 c d-4 \text {$\#$1} b \sqrt {c} d-a^2 f+b^2 d\&,\frac {\text {$\#$1}^2 (-b) f \log \left (-\text {$\#$1}+\sqrt {a+b x+c x^2}-\sqrt {c} x\right )-2 \text {$\#$1} c^{3/2} d \log \left (-\text {$\#$1}+\sqrt {a+b x+c x^2}-\sqrt {c} x\right )+b c d \log \left (-\text {$\#$1}+\sqrt {a+b x+c x^2}-\sqrt {c} x\right )-2 \text {$\#$1} a \sqrt {c} f \log \left (-\text {$\#$1}+\sqrt {a+b x+c x^2}-\sqrt {c} x\right )+2 a b f \log \left (-\text {$\#$1}+\sqrt {a+b x+c x^2}-\sqrt {c} x\right )}{\text {$\#$1}^3 f-\text {$\#$1} a f-2 \text {$\#$1} c d+b \sqrt {c} d}\&\right ]}{2 f \left (a^2 f^2+2 a c d f+b^2 (-d) f+c^2 d^2\right )}-\frac {2 \left (a^3 b f-2 a^3 c f x+a^2 b^2 f x+3 a^2 b c d-2 a^2 c^2 d x-a b^3 d+4 a b^2 c d x+b^4 (-d) x\right )}{c \left (4 a c-b^2\right ) \sqrt {a+b x+c x^2} \left (a^2 f^2+2 a c d f+b^2 (-d) f+c^2 d^2\right )}+\frac {\log \left (-2 c^{3/2} f \sqrt {a+b x+c x^2}+b c f+2 c^2 f x\right )}{c^{3/2} f} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] 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(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 1648, normalized size = 3.54
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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^4}{\left (d-f\,x^2\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \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 \frac {x^{4}}{- a d \sqrt {a + b x + c x^{2}} + a f x^{2} \sqrt {a + b x + c x^{2}} - b d x \sqrt {a + b x + c x^{2}} + b f x^{3} \sqrt {a + b x + c x^{2}} - c d x^{2} \sqrt {a + b x + c x^{2}} + c f x^{4} \sqrt {a + b x + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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