Optimal. Leaf size=205 \[ \frac {3 (4 c d-b e+2 c e x) \sqrt {b x+c x^2}}{4 e^3 (d+e x)}-\frac {\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac {3 \sqrt {c} (2 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{e^4}+\frac {3 \left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{8 \sqrt {d} e^4 \sqrt {c d-b e}} \]
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Rubi [A]
time = 0.13, antiderivative size = 205, 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, 826, 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 {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{8 \sqrt {d} e^4 \sqrt {c d-b e}}-\frac {3 \sqrt {c} (2 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{e^4}+\frac {3 \sqrt {b x+c x^2} (-b e+4 c d+2 c e x)}{4 e^3 (d+e x)}-\frac {\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 634
Rule 738
Rule 746
Rule 826
Rule 857
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx &=-\frac {\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}+\frac {3 \int \frac {(b+2 c x) \sqrt {b x+c x^2}}{(d+e x)^2} \, dx}{4 e}\\ &=\frac {3 (4 c d-b e+2 c e x) \sqrt {b x+c x^2}}{4 e^3 (d+e x)}-\frac {\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac {3 \int \frac {b (4 c d-b e)+4 c (2 c d-b e) x}{(d+e x) \sqrt {b x+c x^2}} \, dx}{8 e^3}\\ &=\frac {3 (4 c d-b e+2 c e x) \sqrt {b x+c x^2}}{4 e^3 (d+e x)}-\frac {\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac {(3 c (2 c d-b e)) \int \frac {1}{\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}{(d+e x) \sqrt {b x+c x^2}} \, dx}{8 e^4}\\ &=\frac {3 (4 c d-b e+2 c e x) \sqrt {b x+c x^2}}{4 e^3 (d+e x)}-\frac {\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac {(3 c (2 c d-b e)) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {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}{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 )}{4 e^4}\\ &=\frac {3 (4 c d-b e+2 c e x) \sqrt {b x+c x^2}}{4 e^3 (d+e x)}-\frac {\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac {3 \sqrt {c} (2 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{e^4}+\frac {3 \left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) \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 )}{8 \sqrt {d} e^4 \sqrt {c d-b e}}\\ \end {align*}
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Mathematica [A]
time = 10.99, size = 259, normalized size = 1.26 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\frac {e \sqrt {x} \left (-b^2 e^2 (3 d+5 e x)-2 c^2 d \left (6 d^2+9 d e x+2 e^2 x^2\right )+b c e \left (15 d^2+23 d e x+4 e^2 x^2\right )\right )}{(d+e x)^2}+\frac {12 \sqrt {c} \left (2 c^2 d^2-3 b c d e+b^2 e^2\right ) \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {1+\frac {c x}{b}}}-\frac {3 \sqrt {c d-b e} \left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )}{\sqrt {d} \sqrt {b+c x}}\right )}{4 e^4 (-c d+b e) \sqrt {x}} \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. \(1627\) vs.
\(2(177)=354\).
time = 0.53, size = 1628, normalized size = 7.94
method | result | size |
default | \(\text {Expression too large to display}\) | \(1628\) |
risch | \(\text {Expression too large to display}\) | \(2570\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 423 vs.
\(2 (181) = 362\).
time = 2.15, size = 1711, normalized size = 8.35 \begin {gather*} \left [-\frac {12 \, {\left (2 \, c^{2} d^{5} + b^{2} d x^{2} e^{4} - {\left (3 \, b c d^{2} x^{2} - 2 \, b^{2} d^{2} x\right )} e^{3} + {\left (2 \, c^{2} d^{3} x^{2} - 6 \, b c d^{3} x + b^{2} d^{3}\right )} e^{2} + {\left (4 \, c^{2} d^{4} x - 3 \, b c d^{4}\right )} e\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 3 \, {\left (8 \, c^{2} d^{4} + b^{2} x^{2} e^{4} - 2 \, {\left (4 \, b c d x^{2} - b^{2} d x\right )} e^{3} + {\left (8 \, c^{2} d^{2} x^{2} - 16 \, b c d^{2} x + b^{2} d^{2}\right )} e^{2} + 8 \, {\left (2 \, c^{2} d^{3} x - b c d^{3}\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^{4} e - {\left (4 \, b c d x^{2} - 5 \, b^{2} d x\right )} e^{4} + {\left (4 \, c^{2} d^{2} x^{2} - 23 \, b c d^{2} x + 3 \, b^{2} d^{2}\right )} e^{3} + 3 \, {\left (6 \, c^{2} d^{3} x - 5 \, b c d^{3}\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{8 \, {\left (c d^{4} e^{4} - b d x^{2} e^{7} + {\left (c d^{2} x^{2} - 2 \, b d^{2} x\right )} e^{6} + {\left (2 \, c d^{3} x - b d^{3}\right )} e^{5}\right )}}, \frac {3 \, {\left (8 \, c^{2} d^{4} + b^{2} x^{2} e^{4} - 2 \, {\left (4 \, b c d x^{2} - b^{2} d x\right )} e^{3} + {\left (8 \, c^{2} d^{2} x^{2} - 16 \, b c d^{2} x + b^{2} d^{2}\right )} e^{2} + 8 \, {\left (2 \, c^{2} d^{3} x - b c d^{3}\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 ) - 6 \, {\left (2 \, c^{2} d^{5} + b^{2} d x^{2} e^{4} - {\left (3 \, b c d^{2} x^{2} - 2 \, b^{2} d^{2} x\right )} e^{3} + {\left (2 \, c^{2} d^{3} x^{2} - 6 \, b c d^{3} x + b^{2} d^{3}\right )} e^{2} + {\left (4 \, c^{2} d^{4} x - 3 \, b c d^{4}\right )} e\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + {\left (12 \, c^{2} d^{4} e - {\left (4 \, b c d x^{2} - 5 \, b^{2} d x\right )} e^{4} + {\left (4 \, c^{2} d^{2} x^{2} - 23 \, b c d^{2} x + 3 \, b^{2} d^{2}\right )} e^{3} + 3 \, {\left (6 \, c^{2} d^{3} x - 5 \, b c d^{3}\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{4 \, {\left (c d^{4} e^{4} - b d x^{2} e^{7} + {\left (c d^{2} x^{2} - 2 \, b d^{2} x\right )} e^{6} + {\left (2 \, c d^{3} x - b d^{3}\right )} e^{5}\right )}}, \frac {24 \, {\left (2 \, c^{2} d^{5} + b^{2} d x^{2} e^{4} - {\left (3 \, b c d^{2} x^{2} - 2 \, b^{2} d^{2} x\right )} e^{3} + {\left (2 \, c^{2} d^{3} x^{2} - 6 \, b c d^{3} x + b^{2} d^{3}\right )} e^{2} + {\left (4 \, c^{2} d^{4} x - 3 \, b c d^{4}\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + 3 \, {\left (8 \, c^{2} d^{4} + b^{2} x^{2} e^{4} - 2 \, {\left (4 \, b c d x^{2} - b^{2} d x\right )} e^{3} + {\left (8 \, c^{2} d^{2} x^{2} - 16 \, b c d^{2} x + b^{2} d^{2}\right )} e^{2} + 8 \, {\left (2 \, c^{2} d^{3} x - b c d^{3}\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^{4} e - {\left (4 \, b c d x^{2} - 5 \, b^{2} d x\right )} e^{4} + {\left (4 \, c^{2} d^{2} x^{2} - 23 \, b c d^{2} x + 3 \, b^{2} d^{2}\right )} e^{3} + 3 \, {\left (6 \, c^{2} d^{3} x - 5 \, b c d^{3}\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{8 \, {\left (c d^{4} e^{4} - b d x^{2} e^{7} + {\left (c d^{2} x^{2} - 2 \, b d^{2} x\right )} e^{6} + {\left (2 \, c d^{3} x - b d^{3}\right )} e^{5}\right )}}, \frac {3 \, {\left (8 \, c^{2} d^{4} + b^{2} x^{2} e^{4} - 2 \, {\left (4 \, b c d x^{2} - b^{2} d x\right )} e^{3} + {\left (8 \, c^{2} d^{2} x^{2} - 16 \, b c d^{2} x + b^{2} d^{2}\right )} e^{2} + 8 \, {\left (2 \, c^{2} d^{3} x - b c d^{3}\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 ) + 12 \, {\left (2 \, c^{2} d^{5} + b^{2} d x^{2} e^{4} - {\left (3 \, b c d^{2} x^{2} - 2 \, b^{2} d^{2} x\right )} e^{3} + {\left (2 \, c^{2} d^{3} x^{2} - 6 \, b c d^{3} x + b^{2} d^{3}\right )} e^{2} + {\left (4 \, c^{2} d^{4} x - 3 \, b c d^{4}\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (12 \, c^{2} d^{4} e - {\left (4 \, b c d x^{2} - 5 \, b^{2} d x\right )} e^{4} + {\left (4 \, c^{2} d^{2} x^{2} - 23 \, b c d^{2} x + 3 \, b^{2} d^{2}\right )} e^{3} + 3 \, {\left (6 \, c^{2} d^{3} x - 5 \, b c d^{3}\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{4 \, {\left (c d^{4} e^{4} - b d x^{2} e^{7} + {\left (c d^{2} x^{2} - 2 \, b d^{2} x\right )} e^{6} + {\left (2 \, c d^{3} x - b d^{3}\right )} e^{5}\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 )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 500 vs.
\(2 (181) = 362\).
time = 2.86, size = 500, normalized size = 2.44 \begin {gather*} \sqrt {c x^{2} + b x} c e^{\left (-3\right )} + \frac {3 \, {\left (8 \, c^{2} d^{2} - 8 \, b c d e + b^{2} e^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e}}\right ) e^{\left (-4\right )}}{4 \, \sqrt {-c d^{2} + b d e}} + \frac {3 \, {\left (2 \, c^{2} d - b c e\right )} e^{\left (-4\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{2 \, \sqrt {c}} + \frac {{\left (24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} c^{2} d^{2} e + 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} c^{\frac {5}{2}} d^{3} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b c^{\frac {3}{2}} d^{2} e + 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b c^{2} d^{3} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b c d e^{2} - 28 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{2} c d^{2} e + 10 \, b^{2} c^{\frac {3}{2}} d^{3} - {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{2} \sqrt {c} d e^{2} - 5 \, b^{3} \sqrt {c} d^{2} e + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{2} e^{3} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{3} d e^{2}\right )} e^{\left (-4\right )}}{4 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} d + b d\right )}^{2}} \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\right )}^{3/2}}{{\left (d+e\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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