Optimal. Leaf size=678 \[ -\frac {2 \sqrt {2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \left (2 c (3 A c d-5 a B e)-3 b c (A e+B d)+4 b^2 B e\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {2 e \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (2 c (3 A c d-5 a B e)-3 b c (A e+B d)+4 b^2 B e\right )}{3 c^2 \left (b^2-4 a c\right )}+\frac {2 (d+e x)^{3/2} \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (b c \left (-29 a B e^2+6 A c d e+3 B c d^2\right )-2 c^2 \left (-9 a A e^2-20 a B d e+3 A c d^2\right )-b^2 c e (6 A e+13 B d)+8 b^3 B e^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt {b^2-4 a c} \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}} \]
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Rubi [A] time = 1.03, antiderivative size = 678, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {818, 832, 843, 718, 424, 419} \[ -\frac {2 \sqrt {2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \left (2 c (3 A c d-5 a B e)-3 b c (A e+B d)+4 b^2 B e\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (b c \left (-29 a B e^2+6 A c d e+3 B c d^2\right )-2 c^2 \left (-9 a A e^2-20 a B d e+3 A c d^2\right )-b^2 c e (6 A e+13 B d)+8 b^3 B e^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt {b^2-4 a c} \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 e \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (2 c (3 A c d-5 a B e)-3 b c (A e+B d)+4 b^2 B e\right )}{3 c^2 \left (b^2-4 a c\right )}+\frac {2 (d+e x)^{3/2} \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 718
Rule 818
Rule 832
Rule 843
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=\frac {2 (d+e x)^{3/2} \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 \int \frac {\sqrt {d+e x} \left (\frac {1}{2} e \left (b^2 B d+3 A b c d-10 a B c d+3 a b B e-6 a A c e\right )+\frac {1}{2} e \left (4 b^2 B e-3 b c (B d+A e)+2 c (3 A c d-5 a B e)\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{c \left (b^2-4 a c\right )}\\ &=\frac {2 (d+e x)^{3/2} \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 e \left (4 b^2 B e-3 b c (B d+A e)+2 c (3 A c d-5 a B e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}+\frac {4 \int \frac {-\frac {1}{4} e \left (4 b^3 B d e-b^2 \left (6 B c d^2+3 A c d e-4 a B e^2\right )+2 a c \left (12 A c d e+5 B \left (3 c d^2-a e^2\right )\right )-b c \left (22 a B d e+3 A \left (c d^2+a e^2\right )\right )\right )-\frac {1}{4} e \left (8 b^3 B e^2-b^2 c e (13 B d+6 A e)-2 c^2 \left (3 A c d^2-20 a B d e-9 a A e^2\right )+b c \left (3 B c d^2+6 A c d e-29 a B e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{3 c^2 \left (b^2-4 a c\right )}\\ &=\frac {2 (d+e x)^{3/2} \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 e \left (4 b^2 B e-3 b c (B d+A e)+2 c (3 A c d-5 a B e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}-\frac {\left (\left (c d^2-b d e+a e^2\right ) \left (4 b^2 B e-3 b c (B d+A e)+2 c (3 A c d-5 a B e)\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{3 c^2 \left (b^2-4 a c\right )}-\frac {\left (8 b^3 B e^2-b^2 c e (13 B d+6 A e)-2 c^2 \left (3 A c d^2-20 a B d e-9 a A e^2\right )+b c \left (3 B c d^2+6 A c d e-29 a B e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{3 c^2 \left (b^2-4 a c\right )}\\ &=\frac {2 (d+e x)^{3/2} \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 e \left (4 b^2 B e-3 b c (B d+A e)+2 c (3 A c d-5 a B e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}-\frac {\left (\sqrt {2} \left (8 b^3 B e^2-b^2 c e (13 B d+6 A e)-2 c^2 \left (3 A c d^2-20 a B d e-9 a A e^2\right )+b c \left (3 B c d^2+6 A c d e-29 a B e^2\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 c^3 \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 {a+b x+c x^2}}-\frac {\left (2 \sqrt {2} \left (c d^2-b d e+a e^2\right ) \left (4 b^2 B e-3 b c (B d+A e)+2 c (3 A c d-5 a B 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^3 \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=\frac {2 (d+e x)^{3/2} \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 e \left (4 b^2 B e-3 b c (B d+A e)+2 c (3 A c d-5 a B e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}-\frac {\sqrt {2} \left (8 b^3 B e^2-b^2 c e (13 B d+6 A e)-2 c^2 \left (3 A c d^2-20 a B d e-9 a A e^2\right )+b c \left (3 B c d^2+6 A c d e-29 a B e^2\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 c^3 \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 {a+b x+c x^2}}-\frac {2 \sqrt {2} \left (c d^2-b d e+a e^2\right ) \left (4 b^2 B e-3 b c (B d+A e)+2 c (3 A c d-5 a B 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^3 \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 12.43, size = 1287, normalized size = 1.90 \[ \frac {\sqrt {d+e x} \left (\frac {2 B e^2}{3 c^2}+\frac {2 \left (-B e^2 x b^3-a B e^2 b^2+A c e^2 x b^2+2 B c d e x b^2+A c^2 d^2 b+a A c e^2 b+2 a B c d e b-B c^2 d^2 x b+3 a B c e^2 x b-2 A c^2 d e x b-2 a B c^2 d^2+2 a^2 B c e^2-4 a A c^2 d e+2 A c^3 d^2 x-2 a A c^2 e^2 x-4 a B c^2 d e x\right )}{c^2 \left (4 a c-b^2\right ) \left (c x^2+b x+a\right )}\right ) \left (c x^2+b x+a\right )^2}{(a+x (b+c x))^{3/2}}+\frac {2 (d+e x)^{3/2} \left (\left (8 B e^2 b^3-c e (13 B d+6 A e) b^2+c \left (3 B c d^2+6 A c e d-29 a B e^2\right ) b+2 c^2 \left (-3 A c d^2+20 a B e d+9 a A e^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 )-\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 (8 B e^2 b^3-c e (13 B d+6 A e) b^2+c \left (3 B c d^2+6 A c e d-29 a B e^2\right ) b+2 c^2 \left (-3 A c d^2+20 a B e d+9 a A e^2\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 (8 B e^3 b^4-e^2 \left (21 c d B+8 \sqrt {\left (b^2-4 a c\right ) e^2} B+6 A c e\right ) b^3+c e \left (6 A e \left (2 c d+\sqrt {\left (b^2-4 a c\right ) e^2}\right )+B \left (15 c d^2+13 \sqrt {\left (b^2-4 a c\right ) e^2} d-37 a e^2\right )\right ) b^2+c \left (a e^2 \left (84 c d B+29 \sqrt {\left (b^2-4 a c\right ) e^2} B+24 A c e\right )-3 c d \sqrt {\left (b^2-4 a c\right ) e^2} (B d+2 A e)\right ) b+2 c^2 \left (10 a^2 B e^3-a \left (10 B d \left (3 c d+2 \sqrt {\left (b^2-4 a c\right ) e^2}\right )+3 A e \left (8 c d+3 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right ) e+3 A c d^2 \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 )}{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}}\right ) \left (c x^2+b x+a\right )^{3/2}}{3 c^3 \left (4 a c-b^2\right ) e (a+x (b+c x))^{3/2} \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 = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B e^{2} x^{3} + A d^{2} + {\left (2 \, B d e + A e^{2}\right )} x^{2} + {\left (B d^{2} + 2 \, A d e\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}}{c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 10385, normalized size = 15.32 \[ \text {output too large to display} \]
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 \[ \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{5/2}}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]
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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