Optimal. Leaf size=797 \[ -\frac {2 \left (a B \left (2 c^2 d-b^2 f+2 a c f\right )+A \left (b^3 f-b c (c d+3 a f)\right )+c \left (A b^2 f+b B (c d-a f)-2 A c (c d+a f)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (3 b^6 B d f^2+24 a^2 B c^2 f (c d+a f)^2-A b^5 f^2 (7 c d+6 a f)-b^4 B f \left (7 c^2 d^2+14 a c d f-3 a^2 f^2\right )+A b^3 c f \left (15 c^2 d^2+46 a c d f+43 a^2 f^2\right )+2 b^2 B c \left (2 c^3 d^3+5 a c^2 d^2 f+4 a^2 c d f^2-11 a^3 f^3\right )-4 A b c^2 \left (2 c^3 d^3+9 a c^2 d^2 f+24 a^2 c d f^2+17 a^3 f^3\right )+c \left (3 b^5 B d f^2-2 A b^4 f^2 (4 c d+3 a f)-8 A c^2 (c d+a f)^2 (2 c d+5 a f)-b^3 B f \left (17 c^2 d^2+10 a c d f-3 a^2 f^2\right )+2 A b^2 c f \left (15 c^2 d^2+22 a c d f+19 a^2 f^2\right )+4 b B c \left (2 c^3 d^3+11 a c^2 d^2 f+4 a^2 c d f^2-5 a^3 f^3\right )\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c^2 d^2+2 a c d f-f \left (b^2 d-a^2 f\right )\right )^2 \sqrt {a+b x+c x^2}}-\frac {\left (B \sqrt {d}-A \sqrt {f}\right ) f^{3/2} \tanh ^{-1}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {d} \left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{5/2}}+\frac {\left (B \sqrt {d}+A \sqrt {f}\right ) f^{3/2} \tanh ^{-1}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {d} \left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{5/2}} \]
[Out]
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Rubi [A]
time = 1.14, antiderivative size = 796, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1032, 1078,
1047, 738, 212} \begin {gather*} -\frac {\left (B \sqrt {d}-A \sqrt {f}\right ) \tanh ^{-1}\left (\frac {-2 \sqrt {f} a+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x+b \sqrt {d}}{2 \sqrt {-\sqrt {d} \sqrt {f} b+c d+a f} \sqrt {c x^2+b x+a}}\right ) f^{3/2}}{2 \sqrt {d} \left (-\sqrt {d} \sqrt {f} b+c d+a f\right )^{5/2}}+\frac {\left (\sqrt {f} A+B \sqrt {d}\right ) \tanh ^{-1}\left (\frac {2 \sqrt {f} a+\left (\sqrt {f} b+2 c \sqrt {d}\right ) x+b \sqrt {d}}{2 \sqrt {\sqrt {d} \sqrt {f} b+c d+a f} \sqrt {c x^2+b x+a}}\right ) f^{3/2}}{2 \sqrt {d} \left (\sqrt {d} \sqrt {f} b+c d+a f\right )^{5/2}}-\frac {2 \left (3 B d f^2 b^6-A f^2 (7 c d+6 a f) b^5-B f \left (7 c^2 d^2+14 a c f d-3 a^2 f^2\right ) b^4+A c f \left (15 c^2 d^2+46 a c f d+43 a^2 f^2\right ) b^3+2 B c \left (2 c^3 d^3+5 a c^2 f d^2+4 a^2 c f^2 d-11 a^3 f^3\right ) b^2-4 A c^2 \left (2 c^3 d^3+9 a c^2 f d^2+24 a^2 c f^2 d+17 a^3 f^3\right ) b+24 a^2 B c^2 f (c d+a f)^2+c \left (3 B d f^2 b^5-2 A f^2 (4 c d+3 a f) b^4-B f \left (17 c^2 d^2+10 a c f d-3 a^2 f^2\right ) b^3+2 A c f \left (15 c^2 d^2+22 a c f d+19 a^2 f^2\right ) b^2+4 B c \left (2 c^3 d^3+11 a c^2 f d^2+4 a^2 c f^2 d-5 a^3 f^3\right ) b-8 A c^2 (c d+a f)^2 (2 c d+5 a f)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c^2 d^2+2 a c f d-f \left (b^2 d-a^2 f\right )\right )^2 \sqrt {c x^2+b x+a}}-\frac {2 \left (A f b^3-A c (c d+3 a f) b+a B \left (-f b^2+2 c^2 d+2 a c f\right )+c \left (A f b^2+B (c d-a f) b-2 A c (c d+a f)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \left (c x^2+b x+a\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 738
Rule 1032
Rule 1047
Rule 1078
Rubi steps
\begin {align*} \int \frac {A+B x}{\left (a+b x+c x^2\right )^{5/2} \left (d-f x^2\right )} \, dx &=-\frac {2 \left (A b^3 f-A b c (c d+3 a f)+a B \left (2 c^2 d-b^2 f+2 a c f\right )+c \left (A b^2 f+b B (c d-a f)-2 A c (c d+a f)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \int \frac {\frac {1}{2} \left (3 b^3 B d f-4 b B c d (c d+2 a f)-A b^2 f (7 c d+3 a f)+4 A c \left (2 c^2 d^2+5 a c d f+3 a^2 f^2\right )\right )+\frac {3}{2} \left (b^2-4 a c\right ) f (A b f-B (c d+a f)) x+2 c f \left (A b^2 f+b B (c d-a f)-2 A c (c d+a f)\right ) x^2}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right )}\\ &=-\frac {2 \left (A b^3 f-A b c (c d+3 a f)+a B \left (2 c^2 d-b^2 f+2 a c f\right )+c \left (A b^2 f+b B (c d-a f)-2 A c (c d+a f)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (3 b^6 B d f^2+24 a^2 B c^2 f (c d+a f)^2-A b^5 f^2 (7 c d+6 a f)-b^4 B f \left (7 c^2 d^2+14 a c d f-3 a^2 f^2\right )+A b^3 c f \left (15 c^2 d^2+46 a c d f+43 a^2 f^2\right )+2 b^2 B c \left (2 c^3 d^3+5 a c^2 d^2 f+4 a^2 c d f^2-11 a^3 f^3\right )-4 A b c^2 \left (2 c^3 d^3+9 a c^2 d^2 f+24 a^2 c d f^2+17 a^3 f^3\right )+c \left (3 b^5 B d f^2-2 A b^4 f^2 (4 c d+3 a f)-8 A c^2 (c d+a f)^2 (2 c d+5 a f)-b^3 B f \left (17 c^2 d^2+10 a c d f-3 a^2 f^2\right )+2 A b^2 c f \left (15 c^2 d^2+22 a c d f+19 a^2 f^2\right )+4 b B c \left (2 c^3 d^3+11 a c^2 d^2 f+4 a^2 c d f^2-5 a^3 f^3\right )\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (b^2 d f-(c d+a f)^2\right )^2 \sqrt {a+b x+c x^2}}+\frac {4 \int \frac {\frac {3}{4} \left (b^2-4 a c\right )^2 f^2 \left (A b^2 d f-2 b B d (c d+a f)+A (c d+a f)^2\right )-\frac {3}{4} \left (b^2-4 a c\right )^2 f^2 \left (2 A b f (c d+a f)-B \left (c^2 d^2+2 a c d f+f \left (b^2 d+a^2 f\right )\right )\right ) x}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx}{3 \left (b^2-4 a c\right )^2 \left (b^2 d f-(c d+a f)^2\right )^2}\\ &=-\frac {2 \left (A b^3 f-A b c (c d+3 a f)+a B \left (2 c^2 d-b^2 f+2 a c f\right )+c \left (A b^2 f+b B (c d-a f)-2 A c (c d+a f)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (3 b^6 B d f^2+24 a^2 B c^2 f (c d+a f)^2-A b^5 f^2 (7 c d+6 a f)-b^4 B f \left (7 c^2 d^2+14 a c d f-3 a^2 f^2\right )+A b^3 c f \left (15 c^2 d^2+46 a c d f+43 a^2 f^2\right )+2 b^2 B c \left (2 c^3 d^3+5 a c^2 d^2 f+4 a^2 c d f^2-11 a^3 f^3\right )-4 A b c^2 \left (2 c^3 d^3+9 a c^2 d^2 f+24 a^2 c d f^2+17 a^3 f^3\right )+c \left (3 b^5 B d f^2-2 A b^4 f^2 (4 c d+3 a f)-8 A c^2 (c d+a f)^2 (2 c d+5 a f)-b^3 B f \left (17 c^2 d^2+10 a c d f-3 a^2 f^2\right )+2 A b^2 c f \left (15 c^2 d^2+22 a c d f+19 a^2 f^2\right )+4 b B c \left (2 c^3 d^3+11 a c^2 d^2 f+4 a^2 c d f^2-5 a^3 f^3\right )\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (b^2 d f-(c d+a f)^2\right )^2 \sqrt {a+b x+c x^2}}+\frac {\left (\left (B \sqrt {d}-A \sqrt {f}\right ) f^2\right ) \int \frac {1}{\left (-\sqrt {d} \sqrt {f}-f x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 \sqrt {d} \left (c d-b \sqrt {d} \sqrt {f}+a f\right )^2}+\frac {\left (\left (B \sqrt {d}+A \sqrt {f}\right ) f^2\right ) \int \frac {1}{\left (\sqrt {d} \sqrt {f}-f x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 \sqrt {d} \left (c d+b \sqrt {d} \sqrt {f}+a f\right )^2}\\ &=-\frac {2 \left (A b^3 f-A b c (c d+3 a f)+a B \left (2 c^2 d-b^2 f+2 a c f\right )+c \left (A b^2 f+b B (c d-a f)-2 A c (c d+a f)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (3 b^6 B d f^2+24 a^2 B c^2 f (c d+a f)^2-A b^5 f^2 (7 c d+6 a f)-b^4 B f \left (7 c^2 d^2+14 a c d f-3 a^2 f^2\right )+A b^3 c f \left (15 c^2 d^2+46 a c d f+43 a^2 f^2\right )+2 b^2 B c \left (2 c^3 d^3+5 a c^2 d^2 f+4 a^2 c d f^2-11 a^3 f^3\right )-4 A b c^2 \left (2 c^3 d^3+9 a c^2 d^2 f+24 a^2 c d f^2+17 a^3 f^3\right )+c \left (3 b^5 B d f^2-2 A b^4 f^2 (4 c d+3 a f)-8 A c^2 (c d+a f)^2 (2 c d+5 a f)-b^3 B f \left (17 c^2 d^2+10 a c d f-3 a^2 f^2\right )+2 A b^2 c f \left (15 c^2 d^2+22 a c d f+19 a^2 f^2\right )+4 b B c \left (2 c^3 d^3+11 a c^2 d^2 f+4 a^2 c d f^2-5 a^3 f^3\right )\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (b^2 d f-(c d+a f)^2\right )^2 \sqrt {a+b x+c x^2}}-\frac {\left (\left (B \sqrt {d}-A \sqrt {f}\right ) f^2\right ) \text {Subst}\left (\int \frac {1}{4 c d f-4 b \sqrt {d} f^{3/2}+4 a f^2-x^2} \, dx,x,\frac {b \sqrt {d} \sqrt {f}-2 a f-\left (-2 c \sqrt {d} \sqrt {f}+b f\right ) x}{\sqrt {a+b x+c x^2}}\right )}{\sqrt {d} \left (c d-b \sqrt {d} \sqrt {f}+a f\right )^2}-\frac {\left (\left (B \sqrt {d}+A \sqrt {f}\right ) f^2\right ) \text {Subst}\left (\int \frac {1}{4 c d f+4 b \sqrt {d} f^{3/2}+4 a f^2-x^2} \, dx,x,\frac {-b \sqrt {d} \sqrt {f}-2 a f-\left (2 c \sqrt {d} \sqrt {f}+b f\right ) x}{\sqrt {a+b x+c x^2}}\right )}{\sqrt {d} \left (c d+b \sqrt {d} \sqrt {f}+a f\right )^2}\\ &=-\frac {2 \left (A b^3 f-A b c (c d+3 a f)+a B \left (2 c^2 d-b^2 f+2 a c f\right )+c \left (A b^2 f+b B (c d-a f)-2 A c (c d+a f)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (3 b^6 B d f^2+24 a^2 B c^2 f (c d+a f)^2-A b^5 f^2 (7 c d+6 a f)-b^4 B f \left (7 c^2 d^2+14 a c d f-3 a^2 f^2\right )+A b^3 c f \left (15 c^2 d^2+46 a c d f+43 a^2 f^2\right )+2 b^2 B c \left (2 c^3 d^3+5 a c^2 d^2 f+4 a^2 c d f^2-11 a^3 f^3\right )-4 A b c^2 \left (2 c^3 d^3+9 a c^2 d^2 f+24 a^2 c d f^2+17 a^3 f^3\right )+c \left (3 b^5 B d f^2-2 A b^4 f^2 (4 c d+3 a f)-8 A c^2 (c d+a f)^2 (2 c d+5 a f)-b^3 B f \left (17 c^2 d^2+10 a c d f-3 a^2 f^2\right )+2 A b^2 c f \left (15 c^2 d^2+22 a c d f+19 a^2 f^2\right )+4 b B c \left (2 c^3 d^3+11 a c^2 d^2 f+4 a^2 c d f^2-5 a^3 f^3\right )\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (b^2 d f-(c d+a f)^2\right )^2 \sqrt {a+b x+c x^2}}-\frac {\left (B \sqrt {d}-A \sqrt {f}\right ) f^{3/2} \tanh ^{-1}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {d} \left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{5/2}}+\frac {\left (B \sqrt {d}+A \sqrt {f}\right ) f^{3/2} \tanh ^{-1}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {d} \left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 12.76, size = 674, normalized size = 0.85 \begin {gather*} \frac {2 \left (\frac {4 c \left (-A b^2 f+b B (-c d+a f)+2 A c (c d+a f)\right ) (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)}}-\frac {3 f \left (b^4 B d f+2 c (c d+a f)^2 (-a B+A c x)+b^3 f (-A (c d+2 a f)+B c d x)+b c (c d+a f) (A c d+5 a A f-3 B c d x+a B f x)-b^2 \left (B \left (c^2 d^2+2 a c d f-a^2 f^2\right )+2 a A c f^2 x\right )\right )}{\left (c^2 d^2+2 a c d f+f \left (-b^2 d+a^2 f\right )\right ) \sqrt {a+x (b+c x)}}+\frac {A \left (b^3 f-b c (c d+3 a f)+b^2 c f x-2 c^2 (c d+a f) x\right )+B \left (2 a^2 c f+b c^2 d x+a \left (2 c^2 d-b^2 f-b c f x\right )\right )}{(a+x (b+c x))^{3/2}}+\frac {3 \left (b^2-4 a c\right ) f^{3/2} \left (\frac {\left (-B \sqrt {d}+A \sqrt {f}\right ) \left (c d+b \sqrt {d} \sqrt {f}+a f\right )^2 \tanh ^{-1}\left (\frac {-2 a \sqrt {f}+2 c \sqrt {d} x+b \left (\sqrt {d}-\sqrt {f} x\right )}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+x (b+c x)}}\right )}{\sqrt {c d-b \sqrt {d} \sqrt {f}+a f}}-\frac {\left (B \sqrt {d}+A \sqrt {f}\right ) \left (c d-b \sqrt {d} \sqrt {f}+a f\right )^2 \tanh ^{-1}\left (\frac {-2 \left (a \sqrt {f}+c \sqrt {d} x\right )-b \left (\sqrt {d}+\sqrt {f} x\right )}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+x (b+c x)}}\right )}{\sqrt {c d+b \sqrt {d} \sqrt {f}+a f}}\right )}{4 \sqrt {d} \left (-b^2 d f+(c d+a f)^2\right )}\right )}{3 \left (b^2-4 a c\right ) \left (-b^2 d f+(c d+a f)^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1767\) vs.
\(2(721)=1442\).
time = 0.14, size = 1768, normalized size = 2.22
method | result | size |
default | \(\text {Expression too large to display}\) | \(1768\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {A+B\,x}{\left (d-f\,x^2\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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