Optimal. Leaf size=163 \[ -\frac {2 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2}}+\frac {d^{3/2} (5 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2}}-\frac {d \sqrt {a+b x} \sqrt {c+d x} (2 b c-3 a d)}{a b^2}+\frac {2 (c+d x)^{3/2} (b c-a d)}{a b \sqrt {a+b x}} \]
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Rubi [A] time = 0.15, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {98, 154, 157, 63, 217, 206, 93, 208} \begin {gather*} -\frac {2 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2}}+\frac {d^{3/2} (5 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2}}-\frac {d \sqrt {a+b x} \sqrt {c+d x} (2 b c-3 a d)}{a b^2}+\frac {2 (c+d x)^{3/2} (b c-a d)}{a b \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 98
Rule 154
Rule 157
Rule 206
Rule 208
Rule 217
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/2}}{x (a+b x)^{3/2}} \, dx &=\frac {2 (b c-a d) (c+d x)^{3/2}}{a b \sqrt {a+b x}}+\frac {2 \int \frac {\sqrt {c+d x} \left (\frac {b c^2}{2}-\frac {1}{2} d (2 b c-3 a d) x\right )}{x \sqrt {a+b x}} \, dx}{a b}\\ &=-\frac {d (2 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{a b^2}+\frac {2 (b c-a d) (c+d x)^{3/2}}{a b \sqrt {a+b x}}+\frac {2 \int \frac {\frac {b^2 c^3}{2}+\frac {1}{4} a d^2 (5 b c-3 a d) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{a b^2}\\ &=-\frac {d (2 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{a b^2}+\frac {2 (b c-a d) (c+d x)^{3/2}}{a b \sqrt {a+b x}}+\frac {c^3 \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{a}+\frac {\left (d^2 (5 b c-3 a d)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 b^2}\\ &=-\frac {d (2 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{a b^2}+\frac {2 (b c-a d) (c+d x)^{3/2}}{a b \sqrt {a+b x}}+\frac {\left (2 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{a}+\frac {\left (d^2 (5 b c-3 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{b^3}\\ &=-\frac {d (2 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{a b^2}+\frac {2 (b c-a d) (c+d x)^{3/2}}{a b \sqrt {a+b x}}-\frac {2 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2}}+\frac {\left (d^2 (5 b c-3 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{b^3}\\ &=-\frac {d (2 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{a b^2}+\frac {2 (b c-a d) (c+d x)^{3/2}}{a b \sqrt {a+b x}}-\frac {2 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2}}+\frac {d^{3/2} (5 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [C] time = 1.47, size = 274, normalized size = 1.68 \begin {gather*} -\frac {2 \left (b c \left (\sqrt {b c-a d} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (b c^{3/2} \sqrt {a+b x} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\sqrt {a} \sqrt {c+d x} (a d-b c)\right )-a^{3/2} d^{3/2} \sqrt {a+b x} \sqrt {c+d x} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )\right )+a^{3/2} d \sqrt {c+d x} (b c-a d)^{3/2} \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};\frac {d (a+b x)}{a d-b c}\right )\right )}{a^{3/2} b^2 \sqrt {a+b x} \sqrt {b c-a d} \sqrt {\frac {b (c+d x)}{b c-a d}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.37, size = 229, normalized size = 1.40 \begin {gather*} -\frac {2 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{a^{3/2}}+\frac {\sqrt {c+d x} \left (\frac {2 a^2 b d^2 (c+d x)}{a+b x}-3 a^2 d^3+\frac {2 b^3 c^2 (c+d x)}{a+b x}-\frac {4 a b^2 c d (c+d x)}{a+b x}+5 a b c d^2-2 b^2 c^2 d\right )}{a b^2 \sqrt {a+b x} \left (\frac {b (c+d x)}{a+b x}-d\right )}+\frac {\left (5 b c d^{3/2}-3 a d^{5/2}\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 6.67, size = 1183, normalized size = 7.26
result too large to display
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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maple [B] time = 0.02, size = 492, normalized size = 3.02 \begin {gather*} -\frac {\sqrt {d x +c}\, \left (3 \sqrt {a c}\, a^{2} b \,d^{3} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-5 \sqrt {a c}\, a \,b^{2} c \,d^{2} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+2 \sqrt {b d}\, b^{3} c^{3} x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+3 \sqrt {a c}\, a^{3} d^{3} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-5 \sqrt {a c}\, a^{2} b c \,d^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+2 \sqrt {b d}\, a \,b^{2} c^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-2 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,d^{2} x -6 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d^{2}+8 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c d -4 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2}\right )}{2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, \sqrt {b x +a}\, a \,b^{2}} \end {gather*}
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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^{5/2}}{x\,{\left (a+b\,x\right )}^{3/2}} \,d x \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 (c + d x\right )^{\frac {5}{2}}}{x \left (a + b x\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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