Optimal. Leaf size=384 \[ \frac {5 \left (b^2-4 a c\right )^{9/4} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+11 b d)+13 b^2 e^2+44 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{7392 \sqrt {2} c^{17/4} (b+2 c x)}-\frac {5 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2} \left (-4 c e (2 a e+11 b d)+13 b^2 e^2+44 c^2 d^2\right )}{3696 c^4}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/4} \left (-4 c e (2 a e+11 b d)+13 b^2 e^2+44 c^2 d^2\right )}{308 c^3}+\frac {13 e \left (a+b x+c x^2\right )^{9/4} (2 c d-b e)}{99 c^2}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{9/4}}{11 c} \]
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Rubi [A] time = 0.54, antiderivative size = 384, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {742, 640, 612, 623, 220} \[ -\frac {5 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2} \left (-4 c e (2 a e+11 b d)+13 b^2 e^2+44 c^2 d^2\right )}{3696 c^4}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/4} \left (-4 c e (2 a e+11 b d)+13 b^2 e^2+44 c^2 d^2\right )}{308 c^3}+\frac {5 \left (b^2-4 a c\right )^{9/4} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+11 b d)+13 b^2 e^2+44 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{7392 \sqrt {2} c^{17/4} (b+2 c x)}+\frac {13 e \left (a+b x+c x^2\right )^{9/4} (2 c d-b e)}{99 c^2}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{9/4}}{11 c} \]
Antiderivative was successfully verified.
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Rule 220
Rule 612
Rule 623
Rule 640
Rule 742
Rubi steps
\begin {align*} \int (d+e x)^2 \left (a+b x+c x^2\right )^{5/4} \, dx &=\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{9/4}}{11 c}+\frac {2 \int \left (\frac {1}{4} \left (22 c d^2-4 e \left (\frac {9 b d}{4}+a e\right )\right )+\frac {13}{4} e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/4} \, dx}{11 c}\\ &=\frac {13 e (2 c d-b e) \left (a+b x+c x^2\right )^{9/4}}{99 c^2}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{9/4}}{11 c}+\frac {\left (-\frac {13}{4} b e (2 c d-b e)+\frac {1}{2} c \left (22 c d^2-4 e \left (\frac {9 b d}{4}+a e\right )\right )\right ) \int \left (a+b x+c x^2\right )^{5/4} \, dx}{11 c^2}\\ &=\frac {\left (44 c^2 d^2+13 b^2 e^2-4 c e (11 b d+2 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{308 c^3}+\frac {13 e (2 c d-b e) \left (a+b x+c x^2\right )^{9/4}}{99 c^2}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{9/4}}{11 c}-\frac {\left (5 \left (b^2-4 a c\right ) \left (44 c^2 d^2+13 b^2 e^2-4 c e (11 b d+2 a e)\right )\right ) \int \sqrt [4]{a+b x+c x^2} \, dx}{1232 c^3}\\ &=-\frac {5 \left (b^2-4 a c\right ) \left (44 c^2 d^2+13 b^2 e^2-4 c e (11 b d+2 a e)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{3696 c^4}+\frac {\left (44 c^2 d^2+13 b^2 e^2-4 c e (11 b d+2 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{308 c^3}+\frac {13 e (2 c d-b e) \left (a+b x+c x^2\right )^{9/4}}{99 c^2}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{9/4}}{11 c}+\frac {\left (5 \left (b^2-4 a c\right )^2 \left (44 c^2 d^2+13 b^2 e^2-4 c e (11 b d+2 a e)\right )\right ) \int \frac {1}{\left (a+b x+c x^2\right )^{3/4}} \, dx}{14784 c^4}\\ &=-\frac {5 \left (b^2-4 a c\right ) \left (44 c^2 d^2+13 b^2 e^2-4 c e (11 b d+2 a e)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{3696 c^4}+\frac {\left (44 c^2 d^2+13 b^2 e^2-4 c e (11 b d+2 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{308 c^3}+\frac {13 e (2 c d-b e) \left (a+b x+c x^2\right )^{9/4}}{99 c^2}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{9/4}}{11 c}+\frac {\left (5 \left (b^2-4 a c\right )^2 \left (44 c^2 d^2+13 b^2 e^2-4 c e (11 b d+2 a e)\right ) \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 )}{3696 c^4 (b+2 c x)}\\ &=-\frac {5 \left (b^2-4 a c\right ) \left (44 c^2 d^2+13 b^2 e^2-4 c e (11 b d+2 a e)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{3696 c^4}+\frac {\left (44 c^2 d^2+13 b^2 e^2-4 c e (11 b d+2 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{308 c^3}+\frac {13 e (2 c d-b e) \left (a+b x+c x^2\right )^{9/4}}{99 c^2}+\frac {2 e (d+e x) \left (a+b x+c x^2\right )^{9/4}}{11 c}+\frac {5 \left (b^2-4 a c\right )^{9/4} \left (44 c^2 d^2+13 b^2 e^2-4 c e (11 b d+2 a e)\right ) \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 )}{7392 \sqrt {2} c^{17/4} (b+2 c x)}\\ \end {align*}
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Mathematica [A] time = 0.57, size = 233, normalized size = 0.61 \[ \frac {2 \left (-\frac {\left (c e (2 a e+11 b d)-\frac {13 b^2 e^2}{4}-11 c^2 d^2\right ) \left (24 c^2 (b+2 c x) (a+x (b+c x))^2-5 \left (b^2-4 a c\right ) \left (2 c (b+2 c x) (a+x (b+c x))-\sqrt {2} \left (b^2-4 a c\right )^{3/2} \left (\frac {c (a+x (b+c x))}{4 a c-b^2}\right )^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\right |2\right )\right )\right )}{336 c^4 (a+x (b+c x))^{3/4}}+\frac {13 e (a+x (b+c x))^{9/4} (2 c d-b e)}{18 c}+e (d+e x) (a+x (b+c x))^{9/4}\right )}{11 c} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.09, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c e^{2} x^{4} + {\left (2 \, c d e + b e^{2}\right )} x^{3} + a d^{2} + {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{2} + {\left (b d^{2} + 2 \, a d e\right )} x\right )} {\left (c x^{2} + b x + a\right )}^{\frac {1}{4}}, x\right ) \]
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 \[ \int {\left (c x^{2} + b x + a\right )}^{\frac {5}{4}} {\left (e x + d\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.52, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{2} \left (c \,x^{2}+b x +a \right )^{\frac {5}{4}}\, dx \]
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 \[ \int {\left (c x^{2} + b x + a\right )}^{\frac {5}{4}} {\left (e x + d\right )}^{2}\,{d x} \]
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 \[ \int {\left (d+e\,x\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{5/4} \,d x \]
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 \[ \int \left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac {5}{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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