Optimal. Leaf size=721 \[ \frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-5 a e)-b^2 e^2+16 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{35 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{35 e^4 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {4 \sqrt {a+b x+c x^2} (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )}{35 e^3 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )^2}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-2 c e (7 b d-5 a e)+b^2 e^2+14 c^2 d^2\right )-c d e (5 b d-4 a e)-b e^2 (2 b d-3 a e)+8 c^2 d^3\right )}{35 e^3 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}} \]
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Rubi [A] time = 0.88, antiderivative size = 721, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {732, 810, 834, 843, 718, 424, 419} \[ -\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-2 c e (7 b d-5 a e)+b^2 e^2+14 c^2 d^2\right )-c d e (5 b d-4 a e)-b e^2 (2 b d-3 a e)+8 c^2 d^3\right )}{35 e^3 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}+\frac {4 \sqrt {a+b x+c x^2} (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )}{35 e^3 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )^2}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-5 a e)-b^2 e^2+16 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{35 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{35 e^4 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 732
Rule 810
Rule 834
Rule 843
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx &=-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}+\frac {3 \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{7/2}} \, dx}{7 e}\\ &=-\frac {2 \left (8 c^2 d^3-c d e (5 b d-4 a e)-b e^2 (2 b d-3 a e)+e \left (14 c^2 d^2+b^2 e^2-2 c e (7 b d-5 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{35 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}-\frac {2 \int \frac {\frac {1}{2} \left (5 b^2 c d e+12 a c^2 d e+2 b^3 e^2-8 b c \left (c d^2+2 a e^2\right )\right )-\frac {1}{2} c \left (16 c^2 d^2-b^2 e^2-4 c e (4 b d-5 a e)\right ) x}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx}{35 e^3 \left (c d^2-b d e+a e^2\right )}\\ &=\frac {4 (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) \sqrt {a+b x+c x^2}}{35 e^3 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {2 \left (8 c^2 d^3-c d e (5 b d-4 a e)-b e^2 (2 b d-3 a e)+e \left (14 c^2 d^2+b^2 e^2-2 c e (7 b d-5 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{35 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}+\frac {4 \int \frac {-\frac {1}{4} c \left (b^3 d e^2-4 a c e \left (c d^2+5 a e^2\right )+4 b c d \left (2 c d^2+5 a e^2\right )-b^2 \left (11 c d^2 e-a e^3\right )\right )-\frac {1}{2} c (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{35 e^3 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {4 (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) \sqrt {a+b x+c x^2}}{35 e^3 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {2 \left (8 c^2 d^3-c d e (5 b d-4 a e)-b e^2 (2 b d-3 a e)+e \left (14 c^2 d^2+b^2 e^2-2 c e (7 b d-5 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{35 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}+\frac {\left (c \left (16 c^2 d^2-b^2 e^2-4 c e (4 b d-5 a e)\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{35 e^4 \left (c d^2-b d e+a e^2\right )}-\frac {\left (2 c (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{35 e^4 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {4 (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) \sqrt {a+b x+c x^2}}{35 e^3 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {2 \left (8 c^2 d^3-c d e (5 b d-4 a e)-b e^2 (2 b d-3 a e)+e \left (14 c^2 d^2+b^2 e^2-2 c e (7 b d-5 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{35 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}-\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 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 )}{35 e^4 \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 (16 c^2 d^2-b^2 e^2-4 c e (4 b d-5 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 )}{35 e^4 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=\frac {4 (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) \sqrt {a+b x+c x^2}}{35 e^3 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {2 \left (8 c^2 d^3-c d e (5 b d-4 a e)-b e^2 (2 b d-3 a e)+e \left (14 c^2 d^2+b^2 e^2-2 c e (7 b d-5 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{35 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 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 )}{35 e^4 \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 (16 c^2 d^2-b^2 e^2-4 c e (4 b d-5 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 )}{35 e^4 \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 = 13.38, size = 5469, normalized size = 7.59 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} \sqrt {e x + d}}{e^{5} x^{5} + 5 \, d e^{4} x^{4} + 10 \, d^{2} e^{3} x^{3} + 10 \, d^{3} e^{2} x^{2} + 5 \, d^{4} e x + d^{5}}, x\right ) \]
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 \[ \int \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.34, size = 25722, normalized size = 35.68 \[ \text {output too large to display} \]
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 \[ \int \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (d+e\,x\right )}^{9/2}} \,d x \]
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 \[ \int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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