Optimal. Leaf size=112 \[ -\frac {(a d+3 b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{5/2}}+\frac {2 c^2 \sqrt {a+b x}}{d^2 \sqrt {c+d x} (b c-a d)}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b d^2} \]
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Rubi [A] time = 0.09, antiderivative size = 112, 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, 80, 63, 217, 206} \begin {gather*} -\frac {(a d+3 b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{5/2}}+\frac {2 c^2 \sqrt {a+b x}}{d^2 \sqrt {c+d x} (b c-a d)}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 80
Rule 89
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx &=\frac {2 c^2 \sqrt {a+b x}}{d^2 (b c-a d) \sqrt {c+d x}}-\frac {2 \int \frac {\frac {1}{2} c (b c-a d)-\frac {1}{2} d (b c-a d) x}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{d^2 (b c-a d)}\\ &=\frac {2 c^2 \sqrt {a+b x}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b d^2}-\frac {(3 b c+a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 b d^2}\\ &=\frac {2 c^2 \sqrt {a+b x}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b d^2}-\frac {(3 b c+a d) \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^2 d^2}\\ &=\frac {2 c^2 \sqrt {a+b x}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b d^2}-\frac {(3 b c+a d) \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^2 d^2}\\ &=\frac {2 c^2 \sqrt {a+b x}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b d^2}-\frac {(3 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 151, normalized size = 1.35 \begin {gather*} \frac {b \sqrt {d} \sqrt {a+b x} (b c (3 c+d x)-a d (c+d x))-\sqrt {b c-a d} \left (-a^2 d^2-2 a b c d+3 b^2 c^2\right ) \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 )}{b^2 d^{5/2} \sqrt {c+d x} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.23, size = 146, normalized size = 1.30 \begin {gather*} \frac {\sqrt {a+b x} \left (a^2 d^2-\frac {2 b c^2 d (a+b x)}{c+d x}-2 a b c d+3 b^2 c^2\right )}{b d^2 \sqrt {c+d x} (b c-a d) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {(-a d-3 b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.52, size = 468, normalized size = 4.18 \begin {gather*} \left [\frac {{\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2} + {\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2} - a^{2} 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^{2} d - a b c d^{2} + {\left (b^{2} c d^{2} - a b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, {\left (b^{3} c^{2} d^{3} - a b^{2} c d^{4} + {\left (b^{3} c d^{4} - a b^{2} d^{5}\right )} x\right )}}, \frac {{\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2} + {\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2} - a^{2} 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^{2} d - a b c d^{2} + {\left (b^{2} c d^{2} - a b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{3} c^{2} d^{3} - a b^{2} c d^{4} + {\left (b^{3} c d^{4} - a b^{2} d^{5}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.24, size = 193, normalized size = 1.72 \begin {gather*} \frac {\sqrt {b x + a} {\left (\frac {{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} {\left (b x + a\right )}}{b^{3} c d^{3} {\left | b \right |} - a b^{2} d^{4} {\left | b \right |}} + \frac {3 \, b^{4} c^{2} d - 2 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}}{b^{3} c d^{3} {\left | b \right |} - a b^{2} d^{4} {\left | b \right |}}\right )}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} + \frac {{\left (3 \, b c + a d\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} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 439, normalized size = 3.92 \begin {gather*} -\frac {\sqrt {b x +a}\, \left (a^{2} 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 )+2 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 )-3 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 )+a^{2} 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 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 )-2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,d^{2} x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b c d x -2 \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 )}{2 \sqrt {b d}\, \left (a d -b c \right ) \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {d x +c}\, b \,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 )}^{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 {x^{2}}{\sqrt {a + b x} \left (c + d x\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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