Optimal. Leaf size=102 \[ \frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{5/2}}-\frac {2 \sqrt {a+b x}}{d^2 \sqrt {c+d x}}-\frac {2 c (a+b x)^{3/2}}{3 d (c+d x)^{3/2} (b c-a d)} \]
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Rubi [A] time = 0.06, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {78, 47, 63, 217, 206} \begin {gather*} -\frac {2 \sqrt {a+b x}}{d^2 \sqrt {c+d x}}+\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{5/2}}-\frac {2 c (a+b x)^{3/2}}{3 d (c+d x)^{3/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 78
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {x \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx &=-\frac {2 c (a+b x)^{3/2}}{3 d (b c-a d) (c+d x)^{3/2}}+\frac {\int \frac {\sqrt {a+b x}}{(c+d x)^{3/2}} \, dx}{d}\\ &=-\frac {2 c (a+b x)^{3/2}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 \sqrt {a+b x}}{d^2 \sqrt {c+d x}}+\frac {b \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{d^2}\\ &=-\frac {2 c (a+b x)^{3/2}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 \sqrt {a+b x}}{d^2 \sqrt {c+d x}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{d^2}\\ &=-\frac {2 c (a+b x)^{3/2}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 \sqrt {a+b x}}{d^2 \sqrt {c+d x}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{d^2}\\ &=-\frac {2 c (a+b x)^{3/2}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 \sqrt {a+b x}}{d^2 \sqrt {c+d x}}+\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.67, size = 164, normalized size = 1.61 \begin {gather*} \frac {2 \sqrt {b c-a d} \sqrt {\frac {b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{d^{5/2} \sqrt {c+d x}}-\frac {2 \left (a^2 (-d) (2 c+3 d x)+a b \left (3 c^2+2 c d x-3 d^2 x^2\right )+b^2 c x (3 c+4 d x)\right )}{3 d^2 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 102, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{5/2}}+\frac {2 \sqrt {a+b x} \left (\frac {c d (a+b x)}{c+d x}-3 a d+3 b c\right )}{3 d^2 \sqrt {c+d x} (a d-b c)} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.73, size = 469, normalized size = 4.60 \begin {gather*} \left [\frac {3 \, {\left (b c^{3} - a c^{2} d + {\left (b c d^{2} - a d^{3}\right )} x^{2} + 2 \, {\left (b c^{2} d - a c d^{2}\right )} x\right )} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (3 \, b c^{2} - 2 \, a c d + {\left (4 \, b c d - 3 \, a d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (b c^{3} d^{2} - a c^{2} d^{3} + {\left (b c d^{4} - a d^{5}\right )} x^{2} + 2 \, {\left (b c^{2} d^{3} - a c d^{4}\right )} x\right )}}, -\frac {3 \, {\left (b c^{3} - a c^{2} d + {\left (b c d^{2} - a d^{3}\right )} x^{2} + 2 \, {\left (b c^{2} d - a c d^{2}\right )} x\right )} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) + 2 \, {\left (3 \, b c^{2} - 2 \, a c d + {\left (4 \, b c d - 3 \, a d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (b c^{3} d^{2} - a c^{2} d^{3} + {\left (b c d^{4} - a d^{5}\right )} x^{2} + 2 \, {\left (b c^{2} d^{3} - a c d^{4}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.46, size = 188, normalized size = 1.84 \begin {gather*} -\frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (4 \, b^{4} c d^{2} {\left | b \right |} - 3 \, a b^{3} d^{3} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{3} c d^{3} - a b^{2} d^{4}} + \frac {3 \, {\left (b^{5} c^{2} d {\left | b \right |} - 2 \, a b^{4} c d^{2} {\left | b \right |} + a^{2} b^{3} d^{3} {\left | b \right |}\right )}}{b^{3} c d^{3} - a b^{2} d^{4}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {2 \, {\left | b \right |} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 442, normalized size = 4.33 \begin {gather*} \frac {\left (3 a b \,d^{3} x^{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 )-3 b^{2} c \,d^{2} x^{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 )+6 a b 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 )-6 b^{2} c^{2} d 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 )+3 a b \,c^{2} d \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 )-3 b^{2} c^{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 )-6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,d^{2} x +8 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b c d x -4 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a c d +6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b \,c^{2}\right ) \sqrt {b x +a}}{3 \sqrt {b d}\, \left (a d -b c \right ) \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \left (d x +c \right )^{\frac {3}{2}} d^{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 {x\,\sqrt {a+b\,x}}{{\left (c+d\,x\right )}^{5/2}} \,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 {x \sqrt {a + b x}}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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