Optimal. Leaf size=605 \[ -\frac {\sqrt {2} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {16 \sqrt {2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {d+e x} \left (-c x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-4 b c \left (a e^2+2 c d^2\right )-4 a c^2 d e+b^3 \left (-e^2\right )+9 b^2 c d e\right )}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \]
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Rubi [A] time = 0.58, antiderivative size = 605, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {736, 822, 843, 718, 424, 419} \[ -\frac {2 \sqrt {d+e x} \left (-c x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-4 b c \left (a e^2+2 c d^2\right )-4 a c^2 d e+9 b^2 c d e+b^3 \left (-e^2\right )\right )}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {2} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {16 \sqrt {2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 736
Rule 822
Rule 843
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \int \frac {-4 c d+\frac {b e}{2}-3 c e x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (9 b^2 c d e-4 a c^2 d e-b^3 e^2-4 b c \left (2 c d^2+a e^2\right )-c \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}-\frac {4 \int \frac {-\frac {1}{4} c e \left (7 b^2 d e+4 a c d e-8 b \left (c d^2+a e^2\right )\right )+\frac {1}{4} c e \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (9 b^2 c d e-4 a c^2 d e-b^3 e^2-4 b c \left (2 c d^2+a e^2\right )-c \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}+\frac {(8 c (2 c d-b e)) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2}-\frac {\left (c \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (9 b^2 c d e-4 a c^2 d e-b^3 e^2-4 b c \left (2 c d^2+a e^2\right )-c \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}-\frac {\left (\sqrt {2} \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\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 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right ) \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 (16 \sqrt {2} (2 c d-b e) \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 \left (b^2-4 a c\right )^{3/2} \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (9 b^2 c d e-4 a c^2 d e-b^3 e^2-4 b c \left (2 c d^2+a e^2\right )-c \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\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 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right ) \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 {16 \sqrt {2} (2 c d-b e) \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 \left (b^2-4 a c\right )^{3/2} \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 11.80, size = 1440, normalized size = 2.38 \[ \frac {\sqrt {d+e x} \left (\frac {2 \left (e^2 b^3-9 c d e b^2+c e^2 x b^2+8 c^2 d^2 b+4 a c e^2 b-16 c^2 d e x b+4 a c^2 d e+16 c^3 d^2 x+12 a c^2 e^2 x\right )}{3 \left (4 a c-b^2\right )^2 \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )}-\frac {2 (b+2 c x)}{3 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^2}\right ) \left (c x^2+b x+a\right )^3}{(a+x (b+c x))^{5/2}}+\frac {(d+e x)^{3/2} \left (-4 \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}}} \left (16 d^2 \left (\frac {d}{d+e x}-1\right )^2 c^3+4 e \left (a e \left (\frac {7 d^2}{(d+e x)^2}-\frac {6 d}{d+e x}+3\right )-4 b d \left (\frac {2 d^2}{(d+e x)^2}-\frac {3 d}{d+e x}+1\right )\right ) c^2+e^2 \left (\left (\frac {17 d^2}{(d+e x)^2}-\frac {18 d}{d+e x}+1\right ) b^2-\frac {4 a e \left (\frac {7 d}{d+e x}-3\right ) b}{d+e x}+\frac {12 a^2 e^2}{(d+e x)^2}\right ) c+\frac {b^2 e^3 \left (-\frac {d b}{d+e x}+b+\frac {a e}{d+e x}\right )}{d+e x}\right )+\frac {i \sqrt {2} \left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (16 c^2 d^2+b^2 e^2+4 c e (3 a e-4 b d)\right ) \sqrt {\frac {-\frac {2 a e^2}{d+e x}+b \left (\frac {2 d}{d+e x}-1\right ) e-2 c d \left (\frac {d}{d+e x}-1\right )+\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}}} \sqrt {\frac {\frac {2 a e^2}{d+e x}+2 c d \left (\frac {d}{d+e x}-1\right )+b \left (e-\frac {2 d e}{d+e x}\right )+\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}}} 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 )}{\sqrt {d+e x}}-\frac {i \sqrt {2} \left (-b^3 e^3+b^2 \left (2 c d+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) e^2+4 b \left (a c e^3-4 c d e \sqrt {\left (b^2-4 a c\right ) e^2}\right )+4 c \left (4 c \sqrt {\left (b^2-4 a c\right ) e^2} d^2+a e^2 \left (3 \sqrt {\left (b^2-4 a c\right ) e^2}-2 c d\right )\right )\right ) \sqrt {\frac {-\frac {2 a e^2}{d+e x}+b \left (\frac {2 d}{d+e x}-1\right ) e-2 c d \left (\frac {d}{d+e x}-1\right )+\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}}} \sqrt {\frac {\frac {2 a e^2}{d+e x}+2 c d \left (\frac {d}{d+e x}-1\right )+b \left (e-\frac {2 d e}{d+e x}\right )+\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}}} 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 )}{\sqrt {d+e x}}\right ) \left (c x^2+b x+a\right )^{5/2}}{6 \left (4 a c-b^2\right )^2 e \left (c d^2-b e d+a e^2\right ) \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}}} (a+x (b+c x))^{5/2} \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}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.12, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x + a} \sqrt {e x + d}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, {\left (b^{2} c + a c^{2}\right )} x^{4} + 3 \, a^{2} b x + {\left (b^{3} + 6 \, a b c\right )} x^{3} + a^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} x^{2}}, x\right ) \]
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 \[ \int \frac {\sqrt {e x + d}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.19, size = 13069, normalized size = 21.60 \[ \text {output too large to display} \]
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 \[ \int \frac {\sqrt {e x + d}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {d+e\,x}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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