Optimal. Leaf size=684 \[ -\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \left (3 C e (b d-a e)-c \left (e (B d-A e)+2 C d^2\right )\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c e^2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (e \left (3 a e (2 C d-B e)-b \left (-2 A e^2-B d e+4 C d^2\right )\right )+c d \left (e (B d-4 A e)+2 C d^2\right )\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 e^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {a+b x+c x^2} \left (C d^2-e (B d-A e)\right )}{3 e (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}+\frac {2 \sqrt {a+b x+c x^2} \left (e \left (3 a e (2 C d-B e)-b \left (-2 A e^2-B d e+4 C d^2\right )\right )+c d \left (e (B d-4 A e)+2 C d^2\right )\right )}{3 e \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )^2} \]
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Rubi [A] time = 1.19, antiderivative size = 680, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1650, 834, 843, 718, 424, 419} \[ \frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \left (-3 C e (b d-a e)+c e (B d-A e)+2 c C d^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c e^2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (3 a e^2 (2 C d-B e)-b e \left (4 C d^2-e (2 A e+B d)\right )+c d e (B d-4 A e)+2 c C d^3\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 e^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {a+b x+c x^2} \left (C d^2-e (B d-A e)\right )}{3 e (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}+\frac {2 \sqrt {a+b x+c x^2} \left (3 a e^2 (2 C d-B e)-b e \left (4 C d^2-e (2 A e+B d)\right )+c d e (B d-4 A e)+2 c C d^3\right )}{3 e \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 834
Rule 843
Rule 1650
Rubi steps
\begin {align*} \int \frac {A+B x+C x^2}{(d+e x)^{5/2} \sqrt {a+b x+c x^2}} \, dx &=-\frac {2 \left (C d^2-e (B d-A e)\right ) \sqrt {a+b x+c x^2}}{3 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}-\frac {2 \int \frac {-\frac {b C d^2-b e (B d+2 A e)+3 e (A c d-a C d+a B e)}{2 e}-\frac {1}{2} \left (B c d-3 b C d+\frac {2 c C d^2}{e}-A c e+3 a C e\right ) x}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx}{3 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {2 \left (C d^2-e (B d-A e)\right ) \sqrt {a+b x+c x^2}}{3 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}+\frac {2 \left (2 c C d^3+c d e (B d-4 A e)+3 a e^2 (2 C d-B e)-b e \left (4 C d^2-e (B d+2 A e)\right )\right ) \sqrt {a+b x+c x^2}}{3 e \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}+\frac {4 \int \frac {\frac {3 b^2 C d^2 e-b d \left (c C d^2+6 a C e^2+c e (2 B d+A e)\right )+e \left (A c \left (3 c d^2-a e^2\right )+a \left (3 a C e^2-c d (C d-4 B e)\right )\right )}{4 e}-\frac {c \left (2 c C d^3+c d e (B d-4 A e)+3 a e^2 (2 C d-B e)-b e \left (4 C d^2-e (B d+2 A e)\right )\right ) x}{4 e}}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{3 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {2 \left (C d^2-e (B d-A e)\right ) \sqrt {a+b x+c x^2}}{3 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}+\frac {2 \left (2 c C d^3+c d e (B d-4 A e)+3 a e^2 (2 C d-B e)-b e \left (4 C d^2-e (B d+2 A e)\right )\right ) \sqrt {a+b x+c x^2}}{3 e \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}+\frac {\left (2 c C d^2-3 C e (b d-a e)+c e (B d-A e)\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{3 e^2 \left (c d^2-b d e+a e^2\right )}-\frac {\left (c \left (2 c C d^3+c d e (B d-4 A e)+3 a e^2 (2 C d-B e)-b e \left (4 C d^2-e (B d+2 A e)\right )\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{3 e^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {2 \left (C d^2-e (B d-A e)\right ) \sqrt {a+b x+c x^2}}{3 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}+\frac {2 \left (2 c C d^3+c d e (B d-4 A e)+3 a e^2 (2 C d-B e)-b e \left (4 C d^2-e (B d+2 A e)\right )\right ) \sqrt {a+b x+c x^2}}{3 e \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \left (2 c C d^3+c d e (B d-4 A e)+3 a e^2 (2 C d-B e)-b e \left (4 C d^2-e (B d+2 A e)\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{3 e^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {a+b x+c x^2}}+\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} \left (2 c C d^2-3 C e (b d-a e)+c e (B d-A e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{3 c e^2 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=-\frac {2 \left (C d^2-e (B d-A e)\right ) \sqrt {a+b x+c x^2}}{3 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}+\frac {2 \left (2 c C d^3+c d e (B d-4 A e)+3 a e^2 (2 C d-B e)-b e \left (4 C d^2-e (B d+2 A e)\right )\right ) \sqrt {a+b x+c x^2}}{3 e \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (2 c C d^3+c d e (B d-4 A e)+3 a e^2 (2 C d-B e)-b e \left (4 C d^2-e (B d+2 A e)\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 e^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (2 c C d^2-3 C e (b d-a e)+c e (B d-A e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c e^2 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 12.16, size = 1194, normalized size = 1.75 \[ \frac {2 \sqrt {c x^2+b x+a} \left (\frac {i \sqrt {1-\frac {2 \left (c d^2+e (a e-b d)\right )}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {2 \left (c d^2+e (a e-b d)\right )}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}+1} \left (\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (c d \left (2 C d^2+e (B d-4 A e)\right )+e \left (-4 b C d^2+b e (B d+2 A e)-3 a e (B e-2 C d)\right )\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b e d+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )-\left (-6 a^2 C e^4-8 a B c d e^3-3 a B \sqrt {\left (b^2-4 a c\right ) e^2} e^3+2 a c C d^2 e^2+2 A c \left (-3 c d^2-2 \sqrt {\left (b^2-4 a c\right ) e^2} d+a e^2\right ) e^2-b^2 \left (2 C d^2+e (B d+2 A e)\right ) e^2+6 a C d \sqrt {\left (b^2-4 a c\right ) e^2} e^2+b \left (2 A \left (3 c d+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) e^2+B \left (3 c d^2+\sqrt {\left (b^2-4 a c\right ) e^2} d+3 a e^2\right ) e+2 C d \left (3 a e^2-2 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right ) e+B c d^2 \sqrt {\left (b^2-4 a c\right ) e^2} e+2 c C d^3 \sqrt {\left (b^2-4 a c\right ) e^2}\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b e d+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )}{2 \sqrt {2} \sqrt {\frac {c d^2+e (a e-b d)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {d+e x}}-\left (2 c C d^3+c e (B d-4 A e) d-3 a e^2 (B e-2 C d)+b e \left (e (B d+2 A e)-4 C d^2\right )\right ) \left (c \left (\frac {d}{d+e x}-1\right )^2+\frac {e \left (-\frac {d b}{d+e x}+b+\frac {a e}{d+e x}\right )}{d+e x}\right )\right ) (d+e x)^{3/2}}{3 e^3 \left (c d^2-b e d+a e^2\right )^2 \sqrt {a+x (b+c x)} \sqrt {\frac {(d+e x)^2 \left (c \left (\frac {d}{d+e x}-1\right )^2+\frac {e \left (-\frac {d b}{d+e x}+b+\frac {a e}{d+e x}\right )}{d+e x}\right )}{e^2}}}+\frac {\left (c x^2+b x+a\right ) \left (-\frac {2 \left (C d^2-B e d+A e^2\right )}{3 e \left (c d^2-b e d+a e^2\right ) (d+e x)^2}-\frac {2 \left (-2 c C d^3-B c e d^2+4 b C e d^2-b B e^2 d+4 A c e^2 d-6 a C e^2 d-2 A b e^3+3 a B e^3\right )}{3 e \left (c d^2-b e d+a e^2\right )^2 (d+e x)}\right ) \sqrt {d+e x}}{\sqrt {a+x (b+c x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C x^{2} + B x + A\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}}{c e^{3} x^{5} + {\left (3 \, c d e^{2} + b e^{3}\right )} x^{4} + a d^{3} + {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} x^{3} + {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} x^{2} + {\left (b d^{3} + 3 \, a d^{2} e\right )} x}, x\right ) \]
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 \[ \int \frac {C x^{2} + B x + A}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 20481, normalized size = 29.94 \[ \text {output too large to display} \]
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 \[ \int \frac {C x^{2} + B x + A}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {C\,x^2+B\,x+A}{{\left (d+e\,x\right )}^{5/2}\,\sqrt {c\,x^2+b\,x+a}} \,d x \]
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 \[ \int \frac {A + B x + C x^{2}}{\left (d + e x\right )^{\frac {5}{2}} \sqrt {a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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