Optimal. Leaf size=459 \[ \frac {\left (128 c^4 d^4-3 b^4 e^4-2 b^2 c e^3 (5 b d-14 a e)-32 c^3 d^2 e (9 b d-8 a e)+8 c^2 e^2 \left (22 b^2 d^2-39 a b d e+16 a^2 e^2\right )-2 c e (2 c d-b e) \left (16 c^2 d^2-3 b^2 e^2-4 c e (4 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{128 c^2 e^5}+\frac {\left (16 c^2 d^2+3 b^2 e^2-2 c e (11 b d-8 a e)-6 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c e^3}+\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e}-\frac {(2 c d-b e) \left (128 c^4 d^4+3 b^4 e^4+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+16 c^2 e^2 \left (7 b^2 d^2-20 a b d e+15 a^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{5/2} e^6}+\frac {\left (c d^2-b d e+a e^2\right )^{5/2} \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^6} \]
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Rubi [A]
time = 0.48, antiderivative size = 459, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {748, 828, 857,
635, 212, 738} \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (8 c^2 e^2 \left (16 a^2 e^2-39 a b d e+22 b^2 d^2\right )-2 c e x (2 c d-b e) \left (-4 c e (4 b d-7 a e)-3 b^2 e^2+16 c^2 d^2\right )-2 b^2 c e^3 (5 b d-14 a e)-32 c^3 d^2 e (9 b d-8 a e)-3 b^4 e^4+128 c^4 d^4\right )}{128 c^2 e^5}-\frac {(2 c d-b e) \left (16 c^2 e^2 \left (15 a^2 e^2-20 a b d e+7 b^2 d^2\right )+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+3 b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{5/2} e^6}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e (11 b d-8 a e)+3 b^2 e^2-6 c e x (2 c d-b e)+16 c^2 d^2\right )}{48 c e^3}+\frac {\left (a e^2-b d e+c d^2\right )^{5/2} \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e^6}+\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 738
Rule 748
Rule 828
Rule 857
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx &=\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e}-\frac {\int \frac {(b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx}{2 e}\\ &=\frac {\left (16 c^2 d^2+3 b^2 e^2-2 c e (11 b d-8 a e)-6 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c e^3}+\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e}+\frac {\int \frac {\left (\frac {1}{2} \left (8 c e (b d-2 a e)^2+2 (2 c d-b e) \left (2 a c d e-b d \left (4 c d-\frac {3 b e}{2}\right )\right )\right )-\frac {1}{2} (2 c d-b e) \left (16 c^2 d^2-3 b^2 e^2-4 c e (4 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{d+e x} \, dx}{16 c e^3}\\ &=\frac {\left (128 c^4 d^4-3 b^4 e^4-2 b^2 c e^3 (5 b d-14 a e)-32 c^3 d^2 e (9 b d-8 a e)+8 c^2 e^2 \left (22 b^2 d^2-39 a b d e+16 a^2 e^2\right )-2 c e (2 c d-b e) \left (16 c^2 d^2-3 b^2 e^2-4 c e (4 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{128 c^2 e^5}+\frac {\left (16 c^2 d^2+3 b^2 e^2-2 c e (11 b d-8 a e)-6 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c e^3}+\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e}-\frac {\int \frac {\frac {1}{4} \left (d (2 c d-b e) \left (4 b c d-b^2 e-4 a c e\right ) \left (16 c^2 d^2-3 b^2 e^2-4 c e (4 b d-7 a e)\right )+4 c e (b d-2 a e) \left (8 c e (b d-2 a e)^2-d (2 c d-b e) \left (8 b c d-3 b^2 e-4 a c e\right )\right )\right )+\frac {1}{4} (2 c d-b e) \left (128 c^4 d^4+3 b^4 e^4+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+16 c^2 e^2 \left (7 b^2 d^2-20 a b d e+15 a^2 e^2\right )\right ) x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{64 c^2 e^5}\\ &=\frac {\left (128 c^4 d^4-3 b^4 e^4-2 b^2 c e^3 (5 b d-14 a e)-32 c^3 d^2 e (9 b d-8 a e)+8 c^2 e^2 \left (22 b^2 d^2-39 a b d e+16 a^2 e^2\right )-2 c e (2 c d-b e) \left (16 c^2 d^2-3 b^2 e^2-4 c e (4 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{128 c^2 e^5}+\frac {\left (16 c^2 d^2+3 b^2 e^2-2 c e (11 b d-8 a e)-6 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c e^3}+\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e}+\frac {\left (c d^2-b d e+a e^2\right )^3 \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{e^6}-\frac {\left ((2 c d-b e) \left (128 c^4 d^4+3 b^4 e^4+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+16 c^2 e^2 \left (7 b^2 d^2-20 a b d e+15 a^2 e^2\right )\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{256 c^2 e^6}\\ &=\frac {\left (128 c^4 d^4-3 b^4 e^4-2 b^2 c e^3 (5 b d-14 a e)-32 c^3 d^2 e (9 b d-8 a e)+8 c^2 e^2 \left (22 b^2 d^2-39 a b d e+16 a^2 e^2\right )-2 c e (2 c d-b e) \left (16 c^2 d^2-3 b^2 e^2-4 c e (4 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{128 c^2 e^5}+\frac {\left (16 c^2 d^2+3 b^2 e^2-2 c e (11 b d-8 a e)-6 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c e^3}+\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e}-\frac {\left (2 \left (c d^2-b d e+a e^2\right )^3\right ) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{e^6}-\frac {\left ((2 c d-b e) \left (128 c^4 d^4+3 b^4 e^4+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+16 c^2 e^2 \left (7 b^2 d^2-20 a b d e+15 a^2 e^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{128 c^2 e^6}\\ &=\frac {\left (128 c^4 d^4-3 b^4 e^4-2 b^2 c e^3 (5 b d-14 a e)-32 c^3 d^2 e (9 b d-8 a e)+8 c^2 e^2 \left (22 b^2 d^2-39 a b d e+16 a^2 e^2\right )-2 c e (2 c d-b e) \left (16 c^2 d^2-3 b^2 e^2-4 c e (4 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{128 c^2 e^5}+\frac {\left (16 c^2 d^2+3 b^2 e^2-2 c e (11 b d-8 a e)-6 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c e^3}+\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e}-\frac {(2 c d-b e) \left (128 c^4 d^4+3 b^4 e^4+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+16 c^2 e^2 \left (7 b^2 d^2-20 a b d e+15 a^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{5/2} e^6}+\frac {\left (c d^2-b d e+a e^2\right )^{5/2} \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^6}\\ \end {align*}
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Mathematica [A]
time = 10.81, size = 440, normalized size = 0.96 \begin {gather*} \frac {(a+x (b+c x))^{5/2}}{5 e}+\frac {(a+x (b+c x))^{3/2} \left (3 b^2 e^2+4 c^2 d (4 d-3 e x)+2 c e (-11 b d+8 a e+3 b e x)\right )}{48 c e^3}-\frac {\frac {2 e \sqrt {a+x (b+c x)} \left (3 b^4 e^4-64 c^4 d^3 (2 d-e x)+2 b^2 c e^3 (5 b d-14 a e+3 b e x)-4 c^2 e^2 \left (32 a^2 e^2+b^2 d (44 d-5 e x)+2 a b e (-39 d+7 e x)\right )+16 c^3 d e (6 b d (3 d-e x)+a e (-16 d+7 e x))\right )}{c^2}+\frac {(2 c d-b e) \left (128 c^4 d^4+3 b^4 e^4+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+16 c^2 e^2 \left (7 b^2 d^2-20 a b d e+15 a^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{c^{5/2}}+256 \left (c d^2+e (-b d+a e)\right )^{5/2} \tanh ^{-1}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{256 e^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. \(1045\) vs.
\(2(429)=858\).
time = 0.88, size = 1046, normalized size = 2.28
method | result | size |
default | \(\frac {\frac {\left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}\right )^{\frac {5}{2}}}{5}+\frac {\left (b e -2 c d \right ) \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right ) \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (\frac {4 c \left (e^{2} a -b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{4 c}+\frac {\left (\frac {4 c \left (e^{2} a -b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 e}+\frac {\left (e^{2} a -b d e +c \,d^{2}\right ) \left (\frac {\left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}{3}+\frac {\left (b e -2 c d \right ) \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{4 c}+\frac {\left (\frac {4 c \left (e^{2} a -b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{2 e}+\frac {\left (e^{2} a -b d e +c \,d^{2}\right ) \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\right )}{2 e \sqrt {c}}-\frac {\left (e^{2} a -b d e +c \,d^{2}\right ) \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{2}}\right )}{e^{2}}}{e}\) | \(1046\) |
risch | \(\text {Expression too large to display}\) | \(2341\) |
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{d + e x}\, dx \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 {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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