Optimal. Leaf size=600 \[ \frac {4 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right )}{21 c^2}-\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{21 c^3 e \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{21 c^3 e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {4}{7} (d+e x)^{5/2} \sqrt {a+b x+c x^2}+\frac {2 (d+e x)^{3/2} \sqrt {a+b x+c x^2} (2 c d-b e)}{7 c} \]
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Rubi [A] time = 0.89, antiderivative size = 600, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {832, 843, 718, 424, 419} \[ \frac {4 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right )}{21 c^2}-\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{21 c^3 e \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{21 c^3 e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {4}{7} (d+e x)^{5/2} \sqrt {a+b x+c x^2}+\frac {2 (d+e x)^{3/2} \sqrt {a+b x+c x^2} (2 c d-b e)}{7 c} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 832
Rule 843
Rubi steps
\begin {align*} \int \frac {(b+2 c x) (d+e x)^{5/2}}{\sqrt {a+b x+c x^2}} \, dx &=\frac {4}{7} (d+e x)^{5/2} \sqrt {a+b x+c x^2}+\frac {2 \int \frac {(d+e x)^{3/2} \left (\frac {5}{2} c (b d-2 a e)+\frac {5}{2} c (2 c d-b e) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{7 c}\\ &=\frac {2 (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{7 c}+\frac {4}{7} (d+e x)^{5/2} \sqrt {a+b x+c x^2}+\frac {4 \int \frac {\sqrt {d+e x} \left (\frac {5}{4} c \left (b^2 d e-16 a c d e+3 b \left (c d^2+a e^2\right )\right )+\frac {5}{2} c \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{35 c^2}\\ &=\frac {4 \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{21 c^2}+\frac {2 (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{7 c}+\frac {4}{7} (d+e x)^{5/2} \sqrt {a+b x+c x^2}+\frac {8 \int \frac {-\frac {5}{8} c \left (4 b^3 d e^2+2 a c e \left (27 c d^2-5 a e^2\right )-b c d \left (3 c d^2+25 a e^2\right )-b^2 \left (9 c d^2 e-4 a e^3\right )\right )+\frac {5}{8} c (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{105 c^3}\\ &=\frac {4 \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{21 c^2}+\frac {2 (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{7 c}+\frac {4}{7} (d+e x)^{5/2} \sqrt {a+b x+c x^2}+\frac {\left ((2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{21 c^2 e}--\frac {\left (8 \left (-\frac {5}{8} c d (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right )-\frac {5}{8} c e \left (4 b^3 d e^2+2 a c e \left (27 c d^2-5 a e^2\right )-b c d \left (3 c d^2+25 a e^2\right )-b^2 \left (9 c d^2 e-4 a e^3\right )\right )\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{105 c^3 e}\\ &=\frac {4 \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{21 c^2}+\frac {2 (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{7 c}+\frac {4}{7} (d+e x)^{5/2} \sqrt {a+b x+c x^2}+\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 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 )}{21 c^3 e \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} \sqrt {b^2-4 a c} \left (-\frac {5}{8} c d (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right )-\frac {5}{8} c e \left (4 b^3 d e^2+2 a c e \left (27 c d^2-5 a e^2\right )-b c d \left (3 c d^2+25 a e^2\right )-b^2 \left (9 c d^2 e-4 a e^3\right )\right )\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 )}{105 c^4 e \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=\frac {4 \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{21 c^2}+\frac {2 (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{7 c}+\frac {4}{7} (d+e x)^{5/2} \sqrt {a+b x+c x^2}+\frac {\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 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 )}{21 c^3 e \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 {4 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (3 c^2 d^2-3 b c d e+2 b^2 e^2-5 a c e^2\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 )}{21 c^3 e \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 12.64, size = 834, normalized size = 1.39 \[ \frac {4 c \sqrt {d+e x} \left (6 \left (3 d^2+3 e x d+e^2 x^2\right ) c^2-e (9 b d+10 a e+3 b e x) c+4 b^2 e^2\right ) (a+x (b+c x))+\frac {4 (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) (a+x (b+c x))}{\sqrt {d+e x}}+\frac {i (d+e x) \sqrt {1-\frac {2 \left (c d^2+e (a e-b d)\right )}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {4 \left (c d^2+e (a e-b d)\right )}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}+2} \left ((b e-2 c d) \left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b e d+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )+\left (8 b^4 e^4-b^3 \left (27 c d+8 \sqrt {\left (b^2-4 a c\right ) e^2}\right ) e^3+b^2 c \left (27 c d^2+19 \sqrt {\left (b^2-4 a c\right ) e^2} d-37 a e^2\right ) e^2+b c \left (a e^2 \left (108 c d+29 \sqrt {\left (b^2-4 a c\right ) e^2}\right )-9 c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}\right ) e+2 c^2 \left (10 a^2 e^4-a d \left (54 c d+29 \sqrt {\left (b^2-4 a c\right ) e^2}\right ) e^2+3 c d^3 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b e d+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )}{e^2 \sqrt {\frac {c d^2+e (a e-b d)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}}{42 c^3 \sqrt {a+x (b+c x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (2 \, c e^{2} x^{3} + b d^{2} + {\left (4 \, c d e + b e^{2}\right )} x^{2} + 2 \, {\left (c d^{2} + b d e\right )} x\right )} \sqrt {e x + d}}{\sqrt {c x^{2} + b x + a}}, x\right ) \]
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 \[ \int \frac {{\left (2 \, c x + b\right )} {\left (e x + d\right )}^{\frac {5}{2}}}{\sqrt {c x^{2} + b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 6517, normalized size = 10.86 \[ \text {output too large to display} \]
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 \[ \int \frac {{\left (2 \, c x + b\right )} {\left (e x + d\right )}^{\frac {5}{2}}}{\sqrt {c x^{2} + b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (b+2\,c\,x\right )\,{\left (d+e\,x\right )}^{5/2}}{\sqrt {c\,x^2+b\,x+a}} \,d x \]
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 \[ \int \frac {\left (b + 2 c x\right ) \left (d + e x\right )^{\frac {5}{2}}}{\sqrt {a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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