Optimal. Leaf size=198 \[ -\frac {3 (4 c d-3 b e-2 c e x) \sqrt {b x+c x^2}}{4 e^3}-\frac {\left (b x+c x^2\right )^{3/2}}{e (d+e x)}+\frac {3 \left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 \sqrt {c} e^4}-\frac {3 \sqrt {d} \sqrt {c d-b e} (2 c d-b e) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{2 e^4} \]
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Rubi [A]
time = 0.16, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {746, 828, 857,
634, 212, 738} \begin {gather*} \frac {3 \left (b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 \sqrt {c} e^4}-\frac {3 \sqrt {d} \sqrt {c d-b e} (2 c d-b e) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{2 e^4}-\frac {3 \sqrt {b x+c x^2} (-3 b e+4 c d-2 c e x)}{4 e^3}-\frac {\left (b x+c x^2\right )^{3/2}}{e (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 634
Rule 738
Rule 746
Rule 828
Rule 857
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^2} \, dx &=-\frac {\left (b x+c x^2\right )^{3/2}}{e (d+e x)}+\frac {3 \int \frac {(b+2 c x) \sqrt {b x+c x^2}}{d+e x} \, dx}{2 e}\\ &=-\frac {3 (4 c d-3 b e-2 c e x) \sqrt {b x+c x^2}}{4 e^3}-\frac {\left (b x+c x^2\right )^{3/2}}{e (d+e x)}-\frac {3 \int \frac {-b c d (4 c d-3 b e)-c \left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) x}{(d+e x) \sqrt {b x+c x^2}} \, dx}{8 c e^3}\\ &=-\frac {3 (4 c d-3 b e-2 c e x) \sqrt {b x+c x^2}}{4 e^3}-\frac {\left (b x+c x^2\right )^{3/2}}{e (d+e x)}-\frac {(3 d (c d-b e) (2 c d-b e)) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{2 e^4}+\frac {\left (3 \left (8 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{8 e^4}\\ &=-\frac {3 (4 c d-3 b e-2 c e x) \sqrt {b x+c x^2}}{4 e^3}-\frac {\left (b x+c x^2\right )^{3/2}}{e (d+e x)}+\frac {(3 d (c d-b e) (2 c d-b e)) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac {-b d-(2 c d-b e) x}{\sqrt {b x+c x^2}}\right )}{e^4}+\frac {\left (3 \left (8 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{4 e^4}\\ &=-\frac {3 (4 c d-3 b e-2 c e x) \sqrt {b x+c x^2}}{4 e^3}-\frac {\left (b x+c x^2\right )^{3/2}}{e (d+e x)}+\frac {3 \left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 \sqrt {c} e^4}-\frac {3 \sqrt {d} \sqrt {c d-b e} (2 c d-b e) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{2 e^4}\\ \end {align*}
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Mathematica [A]
time = 0.77, size = 225, normalized size = 1.14 \begin {gather*} \frac {(x (b+c x))^{3/2} \left (\frac {e \sqrt {x} \left (b e (9 d+5 e x)-2 c \left (6 d^2+3 d e x-e^2 x^2\right )\right )}{(b+c x) (d+e x)}-\frac {12 \sqrt {d} (2 c d-b e) \sqrt {-c d+b e} \tan ^{-1}\left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {-c d+b e}}\right )}{(b+c x)^{3/2}}-\frac {3 \left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )}{\sqrt {c} (b+c x)^{3/2}}\right )}{4 e^4 x^{3/2}} \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. \(965\) vs.
\(2(170)=340\).
time = 0.50, size = 966, normalized size = 4.88
method | result | size |
default | \(\frac {\frac {e^{2} \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 {d \left (b e -c d \right )}{e^{2}}\right )^{\frac {5}{2}}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )}-\frac {3 e \left (b e -2 c d \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 {d \left (b e -c d \right )}{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 {d \left (b e -c d \right )}{e^{2}}}}{4 c}+\frac {\left (-\frac {4 c d \left (b e -c d \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 {d \left (b e -c d \right )}{e^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{2 e}-\frac {d \left (b e -c d \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 {d \left (b e -c d \right )}{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 {d \left (b e -c d \right )}{e^{2}}}\right )}{2 e \sqrt {c}}+\frac {d \left (b e -c d \right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{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 {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{e^{2}}\right )}{2 d \left (b e -c d \right )}-\frac {4 c \,e^{2} \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 {d \left (b e -c d \right )}{e^{2}}\right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (-\frac {4 c d \left (b e -c d \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 {d \left (b e -c d \right )}{e^{2}}}}{4 c}+\frac {\left (-\frac {4 c d \left (b e -c d \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 {d \left (b e -c d \right )}{e^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{d \left (b e -c d \right )}}{e^{2}}\) | \(966\) |
risch | \(\text {Expression too large to display}\) | \(1361\) |
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 [A]
time = 2.19, size = 1007, normalized size = 5.09 \begin {gather*} \left [\frac {3 \, {\left (8 \, c^{2} d^{3} + b^{2} x e^{3} - {\left (8 \, b c d x - b^{2} d\right )} e^{2} + 8 \, {\left (c^{2} d^{2} x - b c d^{2}\right )} e\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 12 \, {\left (2 \, c^{2} d^{2} - b c x e^{2} + {\left (2 \, c^{2} d x - b c d\right )} e\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {2 \, c d x - b x e + b d + 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{x e + d}\right ) - 2 \, {\left (12 \, c^{2} d^{2} e - {\left (2 \, c^{2} x^{2} + 5 \, b c x\right )} e^{3} + 3 \, {\left (2 \, c^{2} d x - 3 \, b c d\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{8 \, {\left (c x e^{5} + c d e^{4}\right )}}, -\frac {24 \, {\left (2 \, c^{2} d^{2} - b c x e^{2} + {\left (2 \, c^{2} d x - b c d\right )} e\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{c d x - b x e}\right ) - 3 \, {\left (8 \, c^{2} d^{3} + b^{2} x e^{3} - {\left (8 \, b c d x - b^{2} d\right )} e^{2} + 8 \, {\left (c^{2} d^{2} x - b c d^{2}\right )} e\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (12 \, c^{2} d^{2} e - {\left (2 \, c^{2} x^{2} + 5 \, b c x\right )} e^{3} + 3 \, {\left (2 \, c^{2} d x - 3 \, b c d\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{8 \, {\left (c x e^{5} + c d e^{4}\right )}}, -\frac {3 \, {\left (8 \, c^{2} d^{3} + b^{2} x e^{3} - {\left (8 \, b c d x - b^{2} d\right )} e^{2} + 8 \, {\left (c^{2} d^{2} x - b c d^{2}\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + 6 \, {\left (2 \, c^{2} d^{2} - b c x e^{2} + {\left (2 \, c^{2} d x - b c d\right )} e\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {2 \, c d x - b x e + b d + 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{x e + d}\right ) + {\left (12 \, c^{2} d^{2} e - {\left (2 \, c^{2} x^{2} + 5 \, b c x\right )} e^{3} + 3 \, {\left (2 \, c^{2} d x - 3 \, b c d\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{4 \, {\left (c x e^{5} + c d e^{4}\right )}}, -\frac {12 \, {\left (2 \, c^{2} d^{2} - b c x e^{2} + {\left (2 \, c^{2} d x - b c d\right )} e\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{c d x - b x e}\right ) + 3 \, {\left (8 \, c^{2} d^{3} + b^{2} x e^{3} - {\left (8 \, b c d x - b^{2} d\right )} e^{2} + 8 \, {\left (c^{2} d^{2} x - b c d^{2}\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (12 \, c^{2} d^{2} e - {\left (2 \, c^{2} x^{2} + 5 \, b c x\right )} e^{3} + 3 \, {\left (2 \, c^{2} d x - 3 \, b c d\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{4 \, {\left (c x e^{5} + c d e^{4}\right )}}\right ] \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 (x \left (b + c x\right )\right )^{\frac {3}{2}}}{\left (d + e x\right )^{2}}\, 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 = -0.01 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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