Optimal. Leaf size=285 \[ -\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{168 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{14 c^2}+\frac {2 e \left (a+b x+c x^2\right )^{9/4}}{9 c}+\frac {5 \left (b^2-4 a c\right )^{9/4} (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{336 \sqrt {2} c^{13/4} (b+2 c x)} \]
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Rubi [A]
time = 0.15, antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {654, 626, 637,
226} \begin {gather*} \frac {5 \left (b^2-4 a c\right )^{9/4} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) (2 c d-b e) F\left (2 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{336 \sqrt {2} c^{13/4} (b+2 c x)}-\frac {5 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e)}{168 c^3}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/4} (2 c d-b e)}{14 c^2}+\frac {2 e \left (a+b x+c x^2\right )^{9/4}}{9 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 626
Rule 637
Rule 654
Rubi steps
\begin {align*} \int (d+e x) \left (a+b x+c x^2\right )^{5/4} \, dx &=\frac {2 e \left (a+b x+c x^2\right )^{9/4}}{9 c}+\frac {(2 c d-b e) \int \left (a+b x+c x^2\right )^{5/4} \, dx}{2 c}\\ &=\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{14 c^2}+\frac {2 e \left (a+b x+c x^2\right )^{9/4}}{9 c}-\frac {\left (5 \left (b^2-4 a c\right ) (2 c d-b e)\right ) \int \sqrt [4]{a+b x+c x^2} \, dx}{56 c^2}\\ &=-\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{168 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{14 c^2}+\frac {2 e \left (a+b x+c x^2\right )^{9/4}}{9 c}+\frac {\left (5 \left (b^2-4 a c\right )^2 (2 c d-b e)\right ) \int \frac {1}{\left (a+b x+c x^2\right )^{3/4}} \, dx}{672 c^3}\\ &=-\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{168 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{14 c^2}+\frac {2 e \left (a+b x+c x^2\right )^{9/4}}{9 c}+\frac {\left (5 \left (b^2-4 a c\right )^2 (2 c d-b e) \sqrt {(b+2 c x)^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{168 c^3 (b+2 c x)}\\ &=-\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{168 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{14 c^2}+\frac {2 e \left (a+b x+c x^2\right )^{9/4}}{9 c}+\frac {5 \left (b^2-4 a c\right )^{9/4} (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{336 \sqrt {2} c^{13/4} (b+2 c x)}\\ \end {align*}
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Mathematica [A]
time = 10.25, size = 175, normalized size = 0.61 \begin {gather*} \frac {2 e (a+x (b+c x))^{9/4}}{9 c}+\frac {(2 c d-b e) \left (24 c^2 (b+2 c x) (a+x (b+c x))^2-5 \left (b^2-4 a c\right ) \left (2 c (b+2 c x) (a+x (b+c x))-\sqrt {2} \left (b^2-4 a c\right )^{3/2} \left (\frac {c (a+x (b+c x))}{-b^2+4 a c}\right )^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\right |2\right )\right )\right )}{336 c^4 (a+x (b+c x))^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (e x +d \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{4}}\, dx\]
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 \left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{4}}\, 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 (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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