Optimal. Leaf size=285 \[ \frac {5 \left (b^2-4 a c\right )^{9/4} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) (2 c d-b e) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{336 \sqrt {2} c^{13/4} (b+2 c x)}-\frac {5 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e)}{168 c^3}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/4} (2 c d-b e)}{14 c^2}+\frac {2 e \left (a+b x+c x^2\right )^{9/4}}{9 c} \]
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Rubi [A] time = 0.21, 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 = {640, 612, 623, 220} \[ -\frac {5 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e)}{168 c^3}+\frac {5 \left (b^2-4 a c\right )^{9/4} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) (2 c d-b e) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{336 \sqrt {2} c^{13/4} (b+2 c x)}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/4} (2 c d-b e)}{14 c^2}+\frac {2 e \left (a+b x+c x^2\right )^{9/4}}{9 c} \]
Antiderivative was successfully verified.
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Rule 220
Rule 612
Rule 623
Rule 640
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 ) \operatorname {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 = 0.34, size = 175, normalized size = 0.61 \[ \frac {(2 c d-b e) \left (24 c^2 (b+2 c x) (a+x (b+c x))^2-5 \left (b^2-4 a c\right ) \left (2 c (b+2 c x) (a+x (b+c x))-\sqrt {2} \left (b^2-4 a c\right )^{3/2} \left (\frac {c (a+x (b+c x))}{4 a c-b^2}\right )^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\right |2\right )\right )\right )}{336 c^4 (a+x (b+c x))^{3/4}}+\frac {2 e (a+x (b+c x))^{9/4}}{9 c} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.34, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c e x^{3} + {\left (c d + b e\right )} x^{2} + a d + {\left (b d + a e\right )} x\right )} {\left (c x^{2} + b x + a\right )}^{\frac {1}{4}}, x\right ) \]
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 \[ \int {\left (c x^{2} + b x + a\right )}^{\frac {5}{4}} {\left (e x + d\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.76, 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 \[ \int {\left (c x^{2} + b x + a\right )}^{\frac {5}{4}} {\left (e x + d\right )}\,{d x} \]
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 \[ \int \left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/4} \,d x \]
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 \[ \int \left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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