Optimal. Leaf size=126 \[ \frac {2 c^2 \sqrt {a+b x}}{3 d^2 (c+d x)^{3/2} (b c-a d)}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{5/2}}-\frac {4 c \sqrt {a+b x} (2 b c-3 a d)}{3 d^2 \sqrt {c+d x} (b c-a d)^2} \]
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Rubi [A] time = 0.09, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {89, 78, 63, 217, 206} \begin {gather*} \frac {2 c^2 \sqrt {a+b x}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac {4 c \sqrt {a+b x} (2 b c-3 a d)}{3 d^2 \sqrt {c+d x} (b c-a d)^2}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 89
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx &=\frac {2 c^2 \sqrt {a+b x}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {2 \int \frac {\frac {1}{2} c (b c-3 a d)-\frac {3}{2} d (b c-a d) x}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 d^2 (b c-a d)}\\ &=\frac {2 c^2 \sqrt {a+b x}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (2 b c-3 a d) \sqrt {a+b x}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{d^2}\\ &=\frac {2 c^2 \sqrt {a+b x}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (2 b c-3 a d) \sqrt {a+b x}}{3 d^2 (b c-a 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 )}{b d^2}\\ &=\frac {2 c^2 \sqrt {a+b x}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (2 b c-3 a d) \sqrt {a+b x}}{3 d^2 (b c-a 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 )}{b d^2}\\ &=\frac {2 c^2 \sqrt {a+b x}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (2 b c-3 a d) \sqrt {a+b x}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 148, normalized size = 1.17 \begin {gather*} \frac {2 \left (-\frac {3 (b c-a d)^{5/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{3/2} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{b^2 \sqrt {d}}-c^2 \sqrt {a+b x}+\frac {2 c \sqrt {a+b x} (c+d x) (2 b c-3 a d)}{b c-a d}\right )}{3 d^2 (c+d x)^{3/2} (a d-b c)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.16, size = 103, normalized size = 0.82 \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{5/2}}-\frac {2 c \sqrt {a+b x} \left (\frac {c d (a+b x)}{c+d x}-6 a d+3 b c\right )}{3 d^2 \sqrt {c+d x} (a d-b c)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.85, size = 670, normalized size = 5.32 \begin {gather*} \left [\frac {3 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )} \sqrt {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 x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (3 \, b^{2} c^{3} d - 5 \, a b c^{2} d^{2} + 2 \, {\left (2 \, b^{2} c^{2} d^{2} - 3 \, a b c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (b^{3} c^{4} d^{3} - 2 \, a b^{2} c^{3} d^{4} + a^{2} b c^{2} d^{5} + {\left (b^{3} c^{2} d^{5} - 2 \, a b^{2} c d^{6} + a^{2} b d^{7}\right )} x^{2} + 2 \, {\left (b^{3} c^{3} d^{4} - 2 \, a b^{2} c^{2} d^{5} + a^{2} b c d^{6}\right )} x\right )}}, -\frac {3 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (3 \, b^{2} c^{3} d - 5 \, a b c^{2} d^{2} + 2 \, {\left (2 \, b^{2} c^{2} d^{2} - 3 \, a b c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (b^{3} c^{4} d^{3} - 2 \, a b^{2} c^{3} d^{4} + a^{2} b c^{2} d^{5} + {\left (b^{3} c^{2} d^{5} - 2 \, a b^{2} c d^{6} + a^{2} b d^{7}\right )} x^{2} + 2 \, {\left (b^{3} c^{3} d^{4} - 2 \, a b^{2} c^{2} d^{5} + a^{2} b c d^{6}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.33, size = 227, normalized size = 1.80 \begin {gather*} -\frac {2 \, \sqrt {b x + a} {\left (\frac {2 \, {\left (2 \, b^{6} c^{2} d^{2} - 3 \, a b^{5} c d^{3}\right )} {\left (b x + a\right )}}{b^{4} c^{2} d^{3} {\left | b \right |} - 2 \, a b^{3} c d^{4} {\left | b \right |} + a^{2} b^{2} d^{5} {\left | b \right |}} + \frac {3 \, {\left (b^{7} c^{3} d - 3 \, a b^{6} c^{2} d^{2} + 2 \, a^{2} b^{5} c d^{3}\right )}}{b^{4} c^{2} d^{3} {\left | b \right |} - 2 \, a b^{3} c d^{4} {\left | b \right |} + a^{2} b^{2} d^{5} {\left | b \right |}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {2 \, b \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} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 604, normalized size = 4.79 \begin {gather*} \frac {\sqrt {b x +a}\, \left (3 a^{2} d^{4} 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^{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^{2} 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^{2} c \,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 )-12 a b \,c^{2} 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^{3} 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^{2} c^{2} 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 )-6 a b \,c^{3} 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^{4} \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 )+12 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a c \,d^{2} x -8 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b \,c^{2} d x +10 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,c^{2} d -6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b \,c^{3}\right )}{3 \sqrt {b d}\, \left (a d -b c \right )^{2} \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^2}{\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^{2}}{\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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