Optimal. Leaf size=189 \[ \frac {\left (-a^2 d^2-6 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2} d^{7/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {a^2 d}{b}+6 a c-\frac {15 b c^2}{d}\right )}{4 d^2 (b c-a d)}+\frac {2 c^2 (a+b x)^{3/2}}{d^2 \sqrt {c+d x} (b c-a d)}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b d^2} \]
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Rubi [A] time = 0.19, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {89, 80, 50, 63, 217, 206} \begin {gather*} \frac {\left (-a^2 d^2-6 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2} d^{7/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {a^2 d}{b}+6 a c-\frac {15 b c^2}{d}\right )}{4 d^2 (b c-a d)}+\frac {2 c^2 (a+b x)^{3/2}}{d^2 \sqrt {c+d x} (b c-a d)}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
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 (a+b x)^{3/2}}{d^2 (b c-a d) \sqrt {c+d x}}-\frac {2 \int \frac {\sqrt {a+b x} \left (\frac {1}{2} c (3 b c-a d)-\frac {1}{2} d (b c-a d) x\right )}{\sqrt {c+d x}} \, dx}{d^2 (b c-a d)}\\ &=\frac {2 c^2 (a+b x)^{3/2}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b d^2}-\frac {\left (15 b^2 c^2-6 a b c d-a^2 d^2\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{4 b d^2 (b c-a d)}\\ &=\frac {2 c^2 (a+b x)^{3/2}}{d^2 (b c-a d) \sqrt {c+d x}}-\frac {\left (15 b^2 c^2-6 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{4 b d^3 (b c-a d)}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b d^2}+\frac {\left (15 b^2 c^2-6 a b c d-a^2 d^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 b d^3}\\ &=\frac {2 c^2 (a+b x)^{3/2}}{d^2 (b c-a d) \sqrt {c+d x}}-\frac {\left (15 b^2 c^2-6 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{4 b d^3 (b c-a d)}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b d^2}+\frac {\left (15 b^2 c^2-6 a b c d-a^2 d^2\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 )}{4 b^2 d^3}\\ &=\frac {2 c^2 (a+b x)^{3/2}}{d^2 (b c-a d) \sqrt {c+d x}}-\frac {\left (15 b^2 c^2-6 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{4 b d^3 (b c-a d)}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b d^2}+\frac {\left (15 b^2 c^2-6 a b c d-a^2 d^2\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 )}{4 b^2 d^3}\\ &=\frac {2 c^2 (a+b x)^{3/2}}{d^2 (b c-a d) \sqrt {c+d x}}-\frac {\left (15 b^2 c^2-6 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{4 b d^3 (b c-a d)}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b d^2}+\frac {\left (15 b^2 c^2-6 a b c d-a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2} d^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.42, size = 193, normalized size = 1.02 \begin {gather*} \frac {\frac {b \sqrt {d} \left (a^2 d (c+d x)+a b \left (-15 c^2-4 c d x+3 d^2 x^2\right )+b^2 x \left (-15 c^2-5 c d x+2 d^2 x^2\right )\right )}{\sqrt {a+b x}}+\frac {\left (a^3 d^3+5 a^2 b c d^2-21 a b^2 c^2 d+15 b^3 c^3\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 )}{\sqrt {b c-a d}}}{4 b^2 d^{7/2} \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.27, size = 225, normalized size = 1.19 \begin {gather*} \frac {\left (-a^2 d^2-6 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2} d^{7/2}}-\frac {\sqrt {a+b x} \left (-\frac {a^2 d^3 (a+b x)}{c+d x}-a^2 b d^2-\frac {25 b^2 c^2 d (a+b x)}{c+d x}-6 a b^2 c d+\frac {8 b c^2 d^2 (a+b x)^2}{(c+d x)^2}+\frac {10 a b c d^2 (a+b x)}{c+d x}+15 b^3 c^2\right )}{4 b d^3 \sqrt {c+d x} \left (b-\frac {d (a+b x)}{c+d x}\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.40, size = 444, normalized size = 2.35 \begin {gather*} \left [-\frac {{\left (15 \, b^{2} c^{3} - 6 \, a b c^{2} d - a^{2} c d^{2} + {\left (15 \, b^{2} c^{2} d - 6 \, 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 (2 \, b^{2} d^{3} x^{2} - 15 \, b^{2} c^{2} d + a b c d^{2} - {\left (5 \, b^{2} c d^{2} - a b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, {\left (b^{2} d^{5} x + b^{2} c d^{4}\right )}}, -\frac {{\left (15 \, b^{2} c^{3} - 6 \, a b c^{2} d - a^{2} c d^{2} + {\left (15 \, b^{2} c^{2} d - 6 \, 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 (2 \, b^{2} d^{3} x^{2} - 15 \, b^{2} c^{2} d + a b c d^{2} - {\left (5 \, b^{2} c d^{2} - a b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, {\left (b^{2} d^{5} x + b^{2} c d^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.41, size = 204, normalized size = 1.08 \begin {gather*} \frac {\sqrt {b x + a} {\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (b x + a\right )}}{d {\left | b \right |}} - \frac {5 \, b^{3} c d^{3} + 3 \, a b^{2} d^{4}}{b^{2} d^{5} {\left | b \right |}}\right )} - \frac {15 \, b^{4} c^{2} d^{2} - 6 \, a b^{3} c d^{3} - a^{2} b^{2} d^{4}}{b^{2} d^{5} {\left | b \right |}}\right )}}{4 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} - \frac {{\left (15 \, b^{2} c^{2} - 6 \, a b c d - a^{2} d^{2}\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 )}{4 \, \sqrt {b d} d^{3} {\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 = 456, normalized size = 2.41 \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 )+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 )-15 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 )+6 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 )-15 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 )-4 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b \,d^{2} x^{2}-2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,d^{2} x +10 \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 +30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b \,c^{2}\right )}{8 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {d x +c}\, b \,d^{3}} \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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