Optimal. Leaf size=523 \[ -\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {4 e \sqrt {a+b x+c x^2} (2 c d-b e)}{3 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )^2}-\frac {2 e \sqrt {a+b x+c x^2}}{3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )} \]
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Rubi [A] time = 0.41, antiderivative size = 523, 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 = {744, 834, 843, 718, 424, 419} \[ -\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {4 e \sqrt {a+b x+c x^2} (2 c d-b e)}{3 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )^2}-\frac {2 e \sqrt {a+b x+c x^2}}{3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 744
Rule 834
Rule 843
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^{5/2} \sqrt {a+b x+c x^2}} \, dx &=-\frac {2 e \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}-\frac {2 \int \frac {\frac {1}{2} (-3 c d+2 b e)+\frac {c e x}{2}}{(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 e \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}-\frac {4 e (2 c d-b e) \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}+\frac {4 \int \frac {\frac {1}{4} c \left (3 c d^2-e (b d+a e)\right )+\frac {1}{2} c e (2 c d-b e) x}{\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 e \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}-\frac {4 e (2 c d-b e) \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}+\frac {(2 c (2 c d-b e)) \int \frac {\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 {c \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{3 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {2 e \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}-\frac {4 e (2 c d-b e) \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}+\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \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 (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} \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 (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=-\frac {2 e \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}-\frac {4 e (2 c d-b e) \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \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 (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} \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 (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 = 6.05, size = 643, normalized size = 1.23 \[ \frac {2 \left (\frac {i (d+e x)^{5/2} \sqrt {1-\frac {2 \left (e (a e-b d)+c d^2\right )}{(d+e x) \left (\sqrt {e^2 \left (b^2-4 a c\right )}-b e+2 c d\right )}} \sqrt {\frac {2 \left (e (a e-b d)+c d^2\right )}{(d+e x) \left (\sqrt {e^2 \left (b^2-4 a c\right )}+b e-2 c d\right )}+1} \left (\left (c \left (2 d \sqrt {e^2 \left (b^2-4 a c\right )}-a e^2-3 b d e\right )+b e \left (b e-\sqrt {e^2 \left (b^2-4 a c\right )}\right )+3 c^2 d^2\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b e d+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )+(b e-2 c d) \left (\sqrt {e^2 \left (b^2-4 a c\right )}-b e+2 c d\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b e d+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )}{\sqrt {2} \sqrt {\frac {e (a e-b d)+c d^2}{\sqrt {e^2 \left (b^2-4 a c\right )}+b e-2 c d}}}+2 e^2 (d+e x) (a+x (b+c x)) (2 c d-b e)+e^2 (a+x (b+c x)) (e (-a e+3 b d+2 b e x)-c d (5 d+4 e x))\right )}{3 e (d+e x)^{3/2} \sqrt {a+x (b+c x)} \left (e (a e-b d)+c d^2\right )^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x + a} \sqrt {e x + d}}{c e^{3} x^{5} + {\left (3 \, c d e^{2} + b e^{3}\right )} x^{4} + a d^{3} + {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} x^{3} + {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} x^{2} + {\left (b d^{3} + 3 \, a d^{2} e\right )} x}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 5822, normalized size = 11.13 \[ \text {output too large to display} \]
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 \[ \int \frac {1}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (d+e\,x\right )}^{5/2}\,\sqrt {c\,x^2+b\,x+a}} \,d x \]
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 \[ \int \frac {1}{\left (d + e x\right )^{\frac {5}{2}} \sqrt {a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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