Optimal. Leaf size=381 \[ -\frac {2 (b d-a e)^{3/2} \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{5/2}}+\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{b^{5/2}}+\frac {16 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{3 b^{5/2}}-\frac {2 (b d-a e)^{3/2} \log (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{5/2}}-\frac {4 (b d-a e)^{3/2} \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{5/2}}-\frac {16 \sqrt {d+e x} (b d-a e)}{3 b^2}+\frac {2 \sqrt {d+e x} (b d-a e) \log (a+b x)}{b^2}+\frac {2 (d+e x)^{3/2} \log (a+b x)}{3 b}-\frac {4 (d+e x)^{3/2}}{9 b} \]
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Rubi [A] time = 1.53, antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 14, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {2411, 2346, 63, 208, 2348, 12, 1587, 6741, 5984, 5918, 2402, 2315, 2319, 50} \[ -\frac {2 (b d-a e)^{3/2} \text {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{5/2}}-\frac {16 \sqrt {d+e x} (b d-a e)}{3 b^2}+\frac {2 \sqrt {d+e x} (b d-a e) \log (a+b x)}{b^2}+\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{b^{5/2}}+\frac {16 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{3 b^{5/2}}-\frac {2 (b d-a e)^{3/2} \log (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{5/2}}-\frac {4 (b d-a e)^{3/2} \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{5/2}}+\frac {2 (d+e x)^{3/2} \log (a+b x)}{3 b}-\frac {4 (d+e x)^{3/2}}{9 b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 50
Rule 63
Rule 208
Rule 1587
Rule 2315
Rule 2319
Rule 2346
Rule 2348
Rule 2402
Rule 2411
Rule 5918
Rule 5984
Rule 6741
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2} \log (a+b x)}{a+b x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (\frac {b d-a e}{b}+\frac {e x}{b}\right )^{3/2} \log (x)}{x} \, dx,x,a+b x\right )}{b}\\ &=\frac {e \operatorname {Subst}\left (\int \sqrt {\frac {b d-a e}{b}+\frac {e x}{b}} \log (x) \, dx,x,a+b x\right )}{b^2}+\frac {(b d-a e) \operatorname {Subst}\left (\int \frac {\sqrt {\frac {b d-a e}{b}+\frac {e x}{b}} \log (x)}{x} \, dx,x,a+b x\right )}{b^2}\\ &=\frac {2 (d+e x)^{3/2} \log (a+b x)}{3 b}-\frac {2 \operatorname {Subst}\left (\int \frac {\left (\frac {b d-a e}{b}+\frac {e x}{b}\right )^{3/2}}{x} \, dx,x,a+b x\right )}{3 b}+\frac {(e (b d-a e)) \operatorname {Subst}\left (\int \frac {\log (x)}{\sqrt {\frac {b d-a e}{b}+\frac {e x}{b}}} \, dx,x,a+b x\right )}{b^3}+\frac {(b d-a e)^2 \operatorname {Subst}\left (\int \frac {\log (x)}{x \sqrt {\frac {b d-a e}{b}+\frac {e x}{b}}} \, dx,x,a+b x\right )}{b^3}\\ &=-\frac {4 (d+e x)^{3/2}}{9 b}+\frac {2 (b d-a e) \sqrt {d+e x} \log (a+b x)}{b^2}+\frac {2 (d+e x)^{3/2} \log (a+b x)}{3 b}-\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{5/2}}-\frac {(2 (b d-a e)) \operatorname {Subst}\left (\int \frac {\sqrt {\frac {b d-a e}{b}+\frac {e x}{b}}}{x} \, dx,x,a+b x\right )}{3 b^2}-\frac {(2 (b d-a e)) \operatorname {Subst}\left (\int \frac {\sqrt {\frac {b d-a e}{b}+\frac {e x}{b}}}{x} \, dx,x,a+b x\right )}{b^2}-\frac {(b d-a e)^2 \operatorname {Subst}\left (\int -\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e x}{b}}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e} x} \, dx,x,a+b x\right )}{b^3}\\ &=-\frac {16 (b d-a e) \sqrt {d+e x}}{3 b^2}-\frac {4 (d+e x)^{3/2}}{9 b}+\frac {2 (b d-a e) \sqrt {d+e x} \log (a+b x)}{b^2}+\frac {2 (d+e x)^{3/2} \log (a+b x)}{3 b}-\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{5/2}}+\frac {\left (2 (b d-a e)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e x}{b}}}{\sqrt {b d-a e}}\right )}{x} \, dx,x,a+b x\right )}{b^{5/2}}-\frac {\left (2 (b d-a e)^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {\frac {b d-a e}{b}+\frac {e x}{b}}} \, dx,x,a+b x\right )}{3 b^3}-\frac {\left (2 (b d-a e)^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {\frac {b d-a e}{b}+\frac {e x}{b}}} \, dx,x,a+b x\right )}{b^3}\\ &=-\frac {16 (b d-a e) \sqrt {d+e x}}{3 b^2}-\frac {4 (d+e x)^{3/2}}{9 b}+\frac {2 (b d-a e) \sqrt {d+e x} \log (a+b x)}{b^2}+\frac {2 (d+e x)^{3/2} \log (a+b x)}{3 b}-\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{5/2}}+\frac {\left (4 (b d-a e)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b d-a e}}\right )}{a e+b \left (-d+x^2\right )} \, dx,x,\sqrt {d+e x}\right )}{b^{3/2}}-\frac {\left (4 (b d-a e)^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {b d-a e}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{3 b^2 e}-\frac {\left (4 (b d-a e)^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {b d-a e}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{b^2 e}\\ &=-\frac {16 (b d-a e) \sqrt {d+e x}}{3 b^2}-\frac {4 (d+e x)^{3/2}}{9 b}+\frac {16 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{3 b^{5/2}}+\frac {2 (b d-a e) \sqrt {d+e x} \log (a+b x)}{b^2}+\frac {2 (d+e x)^{3/2} \log (a+b x)}{3 b}-\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{5/2}}+\frac {\left (4 (b d-a e)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b d-a e}}\right )}{-b d+a e+b x^2} \, dx,x,\sqrt {d+e x}\right )}{b^{3/2}}\\ &=-\frac {16 (b d-a e) \sqrt {d+e x}}{3 b^2}-\frac {4 (d+e x)^{3/2}}{9 b}+\frac {16 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{3 b^{5/2}}+\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{b^{5/2}}+\frac {2 (b d-a e) \sqrt {d+e x} \log (a+b x)}{b^2}+\frac {2 (d+e x)^{3/2} \log (a+b x)}{3 b}-\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{5/2}}-\frac {(4 (b d-a e)) \operatorname {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b d-a e}}\right )}{1-\frac {\sqrt {b} x}{\sqrt {b d-a e}}} \, dx,x,\sqrt {d+e x}\right )}{b^2}\\ &=-\frac {16 (b d-a e) \sqrt {d+e x}}{3 b^2}-\frac {4 (d+e x)^{3/2}}{9 b}+\frac {16 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{3 b^{5/2}}+\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{b^{5/2}}+\frac {2 (b d-a e) \sqrt {d+e x} \log (a+b x)}{b^2}+\frac {2 (d+e x)^{3/2} \log (a+b x)}{3 b}-\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{5/2}}-\frac {4 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{5/2}}+\frac {(4 (b d-a e)) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {\sqrt {b} x}{\sqrt {b d-a e}}}\right )}{1-\frac {b x^2}{b d-a e}} \, dx,x,\sqrt {d+e x}\right )}{b^2}\\ &=-\frac {16 (b d-a e) \sqrt {d+e x}}{3 b^2}-\frac {4 (d+e x)^{3/2}}{9 b}+\frac {16 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{3 b^{5/2}}+\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{b^{5/2}}+\frac {2 (b d-a e) \sqrt {d+e x} \log (a+b x)}{b^2}+\frac {2 (d+e x)^{3/2} \log (a+b x)}{3 b}-\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{5/2}}-\frac {4 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{5/2}}-\frac {\left (4 (b d-a e)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{5/2}}\\ &=-\frac {16 (b d-a e) \sqrt {d+e x}}{3 b^2}-\frac {4 (d+e x)^{3/2}}{9 b}+\frac {16 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{3 b^{5/2}}+\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{b^{5/2}}+\frac {2 (b d-a e) \sqrt {d+e x} \log (a+b x)}{b^2}+\frac {2 (d+e x)^{3/2} \log (a+b x)}{3 b}-\frac {2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{5/2}}-\frac {4 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{5/2}}-\frac {2 (b d-a e)^{3/2} \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 1.10, size = 663, normalized size = 1.74 \[ \frac {12 b^{3/2} (d+e x)^{3/2} \log (a+b x)+36 b^{3/2} d \sqrt {d+e x} \log (a+b x)-18 (b d-a e)^{3/2} \text {Li}_2\left (\frac {1}{2}-\frac {\sqrt {b} \sqrt {d+e x}}{2 \sqrt {b d-a e}}\right )+18 (b d-a e)^{3/2} \text {Li}_2\left (\frac {1}{2} \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}+1\right )\right )+96 a \sqrt {b} e \sqrt {d+e x}-9 (b d-a e)^{3/2} \log ^2\left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right )+9 (b d-a e)^{3/2} \log ^2\left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right )+18 (b d-a e)^{3/2} \log (a+b x) \log \left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right )-18 (b d-a e)^{3/2} \log \left (\frac {1}{2} \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}+1\right )\right ) \log \left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right )-36 a \sqrt {b} e \sqrt {d+e x} \log (a+b x)-18 (b d-a e)^{3/2} \log (a+b x) \log \left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right )+18 (b d-a e)^{3/2} \log \left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right ) \log \left (\frac {1}{2}-\frac {\sqrt {b} \sqrt {d+e x}}{2 \sqrt {b d-a e}}\right )+96 b d \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )-96 a e \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )-8 b^{3/2} (d+e x)^{3/2}-96 b^{3/2} d \sqrt {d+e x}}{18 b^{5/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )}^{\frac {3}{2}} \log \left (b x + a\right )}{b x + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{\frac {3}{2}} \log \left (b x + a\right )}{b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x +d \right )^{\frac {3}{2}} \ln \left (b x +a \right )}{b x +a}\, dx \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {\ln \left (a+b\,x\right )\,{\left (d+e\,x\right )}^{3/2}}{a+b\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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