Optimal. Leaf size=126 \[ -\frac {2 a^2 \sqrt {c+d x}}{3 b^2 (a+b x)^{3/2} (b c-a d)}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2} \sqrt {d}}+\frac {4 a \sqrt {c+d x} (3 b c-2 a d)}{3 b^2 \sqrt {a+b x} (b c-a d)^2} \]
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Rubi [A] time = 0.08, 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} \[ -\frac {2 a^2 \sqrt {c+d x}}{3 b^2 (a+b x)^{3/2} (b c-a d)}+\frac {4 a \sqrt {c+d x} (3 b c-2 a d)}{3 b^2 \sqrt {a+b x} (b c-a d)^2}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2} \sqrt {d}} \]
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}{(a+b x)^{5/2} \sqrt {c+d x}} \, dx &=-\frac {2 a^2 \sqrt {c+d x}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {2 \int \frac {-\frac {1}{2} a (3 b c-a d)+\frac {3}{2} b (b c-a d) x}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx}{3 b^2 (b c-a d)}\\ &=-\frac {2 a^2 \sqrt {c+d x}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {4 a (3 b c-2 a d) \sqrt {c+d x}}{3 b^2 (b c-a d)^2 \sqrt {a+b x}}+\frac {\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{b^2}\\ &=-\frac {2 a^2 \sqrt {c+d x}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {4 a (3 b c-2 a d) \sqrt {c+d x}}{3 b^2 (b c-a d)^2 \sqrt {a+b 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^3}\\ &=-\frac {2 a^2 \sqrt {c+d x}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {4 a (3 b c-2 a d) \sqrt {c+d x}}{3 b^2 (b c-a d)^2 \sqrt {a+b 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^3}\\ &=-\frac {2 a^2 \sqrt {c+d x}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {4 a (3 b c-2 a d) \sqrt {c+d x}}{3 b^2 (b c-a d)^2 \sqrt {a+b x}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2} \sqrt {d}}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 198, normalized size = 1.57 \[ \frac {2 \sqrt {c+d x} \left (\frac {(a+b x) \left (3 b^2 c^2-a^2 d^2\right )}{d (b c-a d)^2}+\frac {a^2}{a d-b c}-\frac {3 (a+b x) \left (\sqrt {b c-a d} \sqrt {\frac {b (c+d x)}{b c-a d}}-\sqrt {d} \sqrt {a+b x} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )\right )}{d \sqrt {b c-a d} \sqrt {\frac {b (c+d x)}{b c-a d}}}\right )}{3 b^2 (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.19, size = 670, normalized size = 5.32 \[ \left [\frac {3 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (5 \, a^{2} b^{2} c d - 3 \, a^{3} b d^{2} + 2 \, {\left (3 \, a b^{3} c d - 2 \, a^{2} b^{2} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (a^{2} b^{5} c^{2} d - 2 \, a^{3} b^{4} c d^{2} + a^{4} b^{3} d^{3} + {\left (b^{7} c^{2} d - 2 \, a b^{6} c d^{2} + a^{2} b^{5} d^{3}\right )} x^{2} + 2 \, {\left (a b^{6} c^{2} d - 2 \, a^{2} b^{5} c d^{2} + a^{3} b^{4} d^{3}\right )} x\right )}}, -\frac {3 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (5 \, a^{2} b^{2} c d - 3 \, a^{3} b d^{2} + 2 \, {\left (3 \, a b^{3} c d - 2 \, a^{2} b^{2} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (a^{2} b^{5} c^{2} d - 2 \, a^{3} b^{4} c d^{2} + a^{4} b^{3} d^{3} + {\left (b^{7} c^{2} d - 2 \, a b^{6} c d^{2} + a^{2} b^{5} d^{3}\right )} x^{2} + 2 \, {\left (a b^{6} c^{2} d - 2 \, a^{2} b^{5} c d^{2} + a^{3} b^{4} d^{3}\right )} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.43, size = 313, normalized size = 2.48 \[ -\frac {\log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{\sqrt {b d} b {\left | b \right |}} + \frac {8 \, {\left (3 \, \sqrt {b d} a b^{4} c^{2} - 5 \, \sqrt {b d} a^{2} b^{3} c d + 2 \, \sqrt {b d} a^{3} b^{2} d^{2} - 6 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{2} c + 3 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b d + 3 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a\right )}}{3 \, {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3} b {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 604, normalized size = 4.79 \[ \frac {\sqrt {d x +c}\, \left (3 a^{2} b^{2} d^{2} x^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-6 a \,b^{3} c d \,x^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+3 b^{4} c^{2} x^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+6 a^{3} b \,d^{2} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-12 a^{2} b^{2} c d x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+6 a \,b^{3} c^{2} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+3 a^{4} d^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-6 a^{3} b c d \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+3 a^{2} b^{2} c^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-8 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b d x +12 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{2} c x -6 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} d +10 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b c \right )}{3 \sqrt {b d}\, \left (a d -b c \right )^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \left (b x +a \right )^{\frac {3}{2}} b^{2}} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {x^2}{{\left (a+b\,x\right )}^{5/2}\,\sqrt {c+d\,x}} \,d x \]
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 \[ \int \frac {x^{2}}{\left (a + b x\right )^{\frac {5}{2}} \sqrt {c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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