Optimal. Leaf size=511 \[ \frac {c \left (8 (5 f g-e h) \left (c g^2+a h^2\right )^2+h \left (12 a^2 f h^4+4 c^2 g^3 (5 f g-e h)+a c h^2 \left (35 f g^2-h (7 e g-3 d h)\right )\right ) x\right ) \sqrt {a+c x^2}}{8 h^5 \left (c g^2+a h^2\right )^2 (g+h x)}+\frac {\left (4 a^2 h^4 (f g-2 e h)-4 c^2 g^4 (5 f g-e h)-a c g h^2 \left (25 f g^2-h (5 e g-9 d h)\right )-3 h \left (4 a^2 f h^4+a c h^2 \left (17 f g^2-h (5 e g-d h)\right )+2 c^2 g^2 \left (5 f g^2-h (e g+d h)\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{24 h^3 \left (c g^2+a h^2\right )^2 (g+h x)^3}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{4 h \left (c g^2+a h^2\right ) (g+h x)^4}-\frac {c^{3/2} (5 f g-e h) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{h^6}-\frac {c \left (12 a^3 f h^6+8 c^3 g^5 (5 f g-e h)+20 a c^2 g^3 h^2 (5 f g-e h)+3 a^2 c h^4 \left (25 f g^2-h (5 e g-d h)\right )\right ) \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{8 h^6 \left (c g^2+a h^2\right )^{5/2}} \]
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Rubi [A]
time = 0.67, antiderivative size = 511, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {1665, 825, 827,
858, 223, 212, 739} \begin {gather*} \frac {\left (a+c x^2\right )^{3/2} \left (-3 x \left (4 a^2 f h^4+a c h^2 \left (17 f g^2-h (5 e g-d h)\right )+2 c^2 \left (5 f g^4-g^2 h (d h+e g)\right )\right )+4 a^2 h^3 (f g-2 e h)-a c g h \left (25 f g^2-h (5 e g-9 d h)\right )-\frac {4 c^2 g^4 (5 f g-e h)}{h}\right )}{24 h^2 (g+h x)^3 \left (a h^2+c g^2\right )^2}+\frac {c \sqrt {a+c x^2} \left (h x \left (12 a^2 f h^4+a c h^2 \left (35 f g^2-h (7 e g-3 d h)\right )+4 c^2 g^3 (5 f g-e h)\right )+8 \left (a h^2+c g^2\right )^2 (5 f g-e h)\right )}{8 h^5 (g+h x) \left (a h^2+c g^2\right )^2}-\frac {c \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right ) \left (12 a^3 f h^6+3 a^2 c h^4 \left (25 f g^2-h (5 e g-d h)\right )+20 a c^2 g^3 h^2 (5 f g-e h)+8 c^3 g^5 (5 f g-e h)\right )}{8 h^6 \left (a h^2+c g^2\right )^{5/2}}-\frac {c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) (5 f g-e h)}{h^6}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{4 h (g+h x)^4 \left (a h^2+c g^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 739
Rule 825
Rule 827
Rule 858
Rule 1665
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^5} \, dx &=-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{4 h \left (c g^2+a h^2\right ) (g+h x)^4}-\frac {\int \frac {\left (-4 (c d g-a f g+a e h)-\left (4 a f h-c \left (e g-\frac {5 f g^2}{h}-d h\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{(g+h x)^4} \, dx}{4 \left (c g^2+a h^2\right )}\\ &=\frac {\left (4 a^2 h^3 (f g-2 e h)-\frac {4 c^2 g^4 (5 f g-e h)}{h}-a c g h \left (25 f g^2-h (5 e g-9 d h)\right )-3 \left (4 a^2 f h^4+a c h^2 \left (17 f g^2-h (5 e g-d h)\right )+2 c^2 \left (5 f g^4-g^2 h (e g+d h)\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{24 h^2 \left (c g^2+a h^2\right )^2 (g+h x)^3}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{4 h \left (c g^2+a h^2\right ) (g+h x)^4}+\frac {\int \frac {\left (-4 a c \left (5 c f g^3-c g h (e g+3 d h)+4 a h^2 (2 f g-e h)\right )+\frac {2 c \left (12 a^2 f h^4+4 c^2 g^3 (5 f g-e h)+a c h^2 \left (35 f g^2-h (7 e g-3 d h)\right )\right ) x}{h}\right ) \sqrt {a+c x^2}}{(g+h x)^2} \, dx}{16 h^2 \left (c g^2+a h^2\right )^2}\\ &=\frac {c \left (8 (5 f g-e h) \left (c g^2+a h^2\right )^2+h \left (12 a^2 f h^4+4 c^2 g^3 (5 f g-e h)+a c h^2 \left (35 f g^2-h (7 e g-3 d h)\right )\right ) x\right ) \sqrt {a+c x^2}}{8 h^5 \left (c g^2+a h^2\right )^2 (g+h x)}+\frac {\left (4 a^2 h^3 (f g-2 e h)-\frac {4 c^2 g^4 (5 f g-e h)}{h}-a c g h \left (25 f g^2-h (5 e g-9 d h)\right )-3 \left (4 a^2 f h^4+a c h^2 \left (17 f g^2-h (5 e g-d h)\right )+2 c^2 \left (5 f g^4-g^2 h (e g+d h)\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{24 h^2 \left (c g^2+a h^2\right )^2 (g+h x)^3}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{4 h \left (c g^2+a h^2\right ) (g+h x)^4}-\frac {\int \frac {-4 a c \left (12 a^2 f h^4+4 c^2 g^3 (5 f g-e h)+a c h^2 \left (35 f g^2-h (7 e g-3 d h)\right )\right )+\frac {32 c^2 (5 f g-e h) \left (c g^2+a h^2\right )^2 x}{h}}{(g+h x) \sqrt {a+c x^2}} \, dx}{32 h^4 \left (c g^2+a h^2\right )^2}\\ &=\frac {c \left (8 (5 f g-e h) \left (c g^2+a h^2\right )^2+h \left (12 a^2 f h^4+4 c^2 g^3 (5 f g-e h)+a c h^2 \left (35 f g^2-h (7 e g-3 d h)\right )\right ) x\right ) \sqrt {a+c x^2}}{8 h^5 \left (c g^2+a h^2\right )^2 (g+h x)}+\frac {\left (4 a^2 h^3 (f g-2 e h)-\frac {4 c^2 g^4 (5 f g-e h)}{h}-a c g h \left (25 f g^2-h (5 e g-9 d h)\right )-3 \left (4 a^2 f h^4+a c h^2 \left (17 f g^2-h (5 e g-d h)\right )+2 c^2 \left (5 f g^4-g^2 h (e g+d h)\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{24 h^2 \left (c g^2+a h^2\right )^2 (g+h x)^3}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{4 h \left (c g^2+a h^2\right ) (g+h x)^4}-\frac {\left (c^2 (5 f g-e h)\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{h^6}+\frac {\left (c \left (12 a^3 f h^6+8 c^3 g^5 (5 f g-e h)+20 a c^2 g^3 h^2 (5 f g-e h)+3 a^2 c h^4 \left (25 f g^2-h (5 e g-d h)\right )\right )\right ) \int \frac {1}{(g+h x) \sqrt {a+c x^2}} \, dx}{8 h^6 \left (c g^2+a h^2\right )^2}\\ &=\frac {c \left (8 (5 f g-e h) \left (c g^2+a h^2\right )^2+h \left (12 a^2 f h^4+4 c^2 g^3 (5 f g-e h)+a c h^2 \left (35 f g^2-h (7 e g-3 d h)\right )\right ) x\right ) \sqrt {a+c x^2}}{8 h^5 \left (c g^2+a h^2\right )^2 (g+h x)}+\frac {\left (4 a^2 h^3 (f g-2 e h)-\frac {4 c^2 g^4 (5 f g-e h)}{h}-a c g h \left (25 f g^2-h (5 e g-9 d h)\right )-3 \left (4 a^2 f h^4+a c h^2 \left (17 f g^2-h (5 e g-d h)\right )+2 c^2 \left (5 f g^4-g^2 h (e g+d h)\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{24 h^2 \left (c g^2+a h^2\right )^2 (g+h x)^3}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{4 h \left (c g^2+a h^2\right ) (g+h x)^4}-\frac {\left (c^2 (5 f g-e h)\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{h^6}-\frac {\left (c \left (12 a^3 f h^6+8 c^3 g^5 (5 f g-e h)+20 a c^2 g^3 h^2 (5 f g-e h)+3 a^2 c h^4 \left (25 f g^2-h (5 e g-d h)\right )\right )\right ) \text {Subst}\left (\int \frac {1}{c g^2+a h^2-x^2} \, dx,x,\frac {a h-c g x}{\sqrt {a+c x^2}}\right )}{8 h^6 \left (c g^2+a h^2\right )^2}\\ &=\frac {c \left (8 (5 f g-e h) \left (c g^2+a h^2\right )^2+h \left (12 a^2 f h^4+4 c^2 g^3 (5 f g-e h)+a c h^2 \left (35 f g^2-h (7 e g-3 d h)\right )\right ) x\right ) \sqrt {a+c x^2}}{8 h^5 \left (c g^2+a h^2\right )^2 (g+h x)}+\frac {\left (4 a^2 h^3 (f g-2 e h)-\frac {4 c^2 g^4 (5 f g-e h)}{h}-a c g h \left (25 f g^2-h (5 e g-9 d h)\right )-3 \left (4 a^2 f h^4+a c h^2 \left (17 f g^2-h (5 e g-d h)\right )+2 c^2 \left (5 f g^4-g^2 h (e g+d h)\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{24 h^2 \left (c g^2+a h^2\right )^2 (g+h x)^3}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{4 h \left (c g^2+a h^2\right ) (g+h x)^4}-\frac {c^{3/2} (5 f g-e h) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{h^6}-\frac {c \left (12 a^3 f h^6+8 c^3 g^5 (5 f g-e h)+20 a c^2 g^3 h^2 (5 f g-e h)+3 a^2 c h^4 \left (25 f g^2-h (5 e g-d h)\right )\right ) \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{8 h^6 \left (c g^2+a h^2\right )^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 11.40, size = 575, normalized size = 1.13 \begin {gather*} -\frac {\frac {h \sqrt {a+c x^2} \left (6 \left (c g^2+a h^2\right )^3 \left (f g^2+h (-e g+d h)\right )-2 \left (c g^2+a h^2\right )^2 \left (17 c f g^3+c g h (-13 e g+9 d h)-4 a h^2 (-2 f g+e h)\right ) (g+h x)+\left (c g^2+a h^2\right ) \left (12 a^2 f h^4+2 c^2 \left (43 f g^4+g^2 h (-23 e g+9 d h)\right )+a c h^2 \left (95 f g^2+h (-43 e g+15 d h)\right )\right ) (g+h x)^2-c \left (4 a^2 h^4 (31 f g-8 e h)+2 c^2 \left (77 f g^5+g^3 h (-25 e g+3 d h)\right )+a c g h^2 \left (287 f g^2+h (-91 e g+15 d h)\right )\right ) (g+h x)^3-24 c f \left (c g^2+a h^2\right )^2 (g+h x)^4\right )}{\left (c g^2+a h^2\right )^2 (g+h x)^4}-\frac {3 c \left (12 a^3 f h^6+8 c^3 g^5 (5 f g-e h)+20 a c^2 g^3 h^2 (5 f g-e h)+3 a^2 c h^4 \left (25 f g^2+h (-5 e g+d h)\right )\right ) \log (g+h x)}{\left (c g^2+a h^2\right )^{5/2}}+24 c^{3/2} (5 f g-e h) \log \left (c x+\sqrt {c} \sqrt {a+c x^2}\right )+\frac {3 c \left (12 a^3 f h^6+8 c^3 g^5 (5 f g-e h)+20 a c^2 g^3 h^2 (5 f g-e h)+3 a^2 c h^4 \left (25 f g^2+h (-5 e g+d h)\right )\right ) \log \left (a h-c g x+\sqrt {c g^2+a h^2} \sqrt {a+c x^2}\right )}{\left (c g^2+a h^2\right )^{5/2}}}{24 h^6} \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. \(7960\) vs.
\(2(485)=970\).
time = 0.14, size = 7961, normalized size = 15.58
method | result | size |
default | \(\text {Expression too large to display}\) | \(7961\) |
risch | \(\text {Expression too large to display}\) | \(11887\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 4346 vs.
\(2 (492) = 984\).
time = 0.51, size = 4346, normalized size = 8.50 \begin {gather*} \text {Too large to display} \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x + f x^{2}\right )}{\left (g + h x\right )^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right )}{{\left (g+h\,x\right )}^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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