Optimal. Leaf size=165 \[ \frac {2 c^2 (a+b x)^{3/2}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac {(5 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{7/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} (5 b c-a d)}{d^3 (b c-a d)}-\frac {4 c (a+b x)^{3/2}}{d^2 \sqrt {c+d x} (b c-a d)} \]
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Rubi [A] time = 0.16, antiderivative size = 165, 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, 78, 50, 63, 217, 206} \begin {gather*} \frac {2 c^2 (a+b x)^{3/2}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac {4 c (a+b x)^{3/2}}{d^2 \sqrt {c+d x} (b c-a d)}+\frac {\sqrt {a+b x} \sqrt {c+d x} (5 b c-a d)}{d^3 (b c-a d)}-\frac {(5 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
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 (a+b x)^{3/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {2 \int \frac {\sqrt {a+b x} \left (\frac {3}{2} c (b c-a d)-\frac {3}{2} d (b c-a d) x\right )}{(c+d x)^{3/2}} \, dx}{3 d^2 (b c-a d)}\\ &=\frac {2 c^2 (a+b x)^{3/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (a+b x)^{3/2}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {(5 b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{d^2 (b c-a d)}\\ &=\frac {2 c^2 (a+b x)^{3/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (a+b x)^{3/2}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {(5 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{d^3 (b c-a d)}-\frac {(5 b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 d^3}\\ &=\frac {2 c^2 (a+b x)^{3/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (a+b x)^{3/2}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {(5 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{d^3 (b c-a d)}-\frac {(5 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 d^3}\\ &=\frac {2 c^2 (a+b x)^{3/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (a+b x)^{3/2}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {(5 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{d^3 (b c-a d)}-\frac {(5 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 d^3}\\ &=\frac {2 c^2 (a+b x)^{3/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (a+b x)^{3/2}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {(5 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{d^3 (b c-a d)}-\frac {(5 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.79, size = 203, normalized size = 1.23 \begin {gather*} \frac {\frac {\sqrt {d} \left (a^2 (-d) \left (13 c^2+18 c d x+3 d^2 x^2\right )+a b \left (15 c^3+7 c^2 d x-15 c d^2 x^2-3 d^3 x^3\right )+b^2 c x \left (15 c^2+20 c d x+3 d^2 x^2\right )\right )}{\sqrt {a+b x} (b c-a d)}-\frac {3 (b c-a d)^{3/2} (5 b c-a d) \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}}{3 d^{7/2} (c+d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.26, size = 188, normalized size = 1.14 \begin {gather*} \frac {(a d-5 b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{7/2}}-\frac {\sqrt {a+b x} \left (-3 a^2 d^2+\frac {2 c^2 d^2 (a+b x)^2}{(c+d x)^2}+\frac {10 b c^2 d (a+b x)}{c+d x}-\frac {12 a c d^2 (a+b x)}{c+d x}+18 a b c d-15 b^2 c^2\right )}{3 d^3 \sqrt {c+d x} (a d-b c) \left (\frac {d (a+b x)}{c+d x}-b\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.89, size = 640, normalized size = 3.88 \begin {gather*} \left [-\frac {3 \, {\left (5 \, b^{2} c^{4} - 6 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (5 \, b^{2} c^{2} d^{2} - 6 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (5 \, b^{2} c^{3} d - 6 \, 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 (15 \, b^{2} c^{3} d - 13 \, a b c^{2} d^{2} + 3 \, {\left (b^{2} c d^{3} - a b d^{4}\right )} x^{2} + 2 \, {\left (10 \, b^{2} c^{2} d^{2} - 9 \, a b c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{12 \, {\left (b^{2} c^{3} d^{4} - a b c^{2} d^{5} + {\left (b^{2} c d^{6} - a b d^{7}\right )} x^{2} + 2 \, {\left (b^{2} c^{2} d^{5} - a b c d^{6}\right )} x\right )}}, \frac {3 \, {\left (5 \, b^{2} c^{4} - 6 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (5 \, b^{2} c^{2} d^{2} - 6 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (5 \, b^{2} c^{3} d - 6 \, 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 (15 \, b^{2} c^{3} d - 13 \, a b c^{2} d^{2} + 3 \, {\left (b^{2} c d^{3} - a b d^{4}\right )} x^{2} + 2 \, {\left (10 \, b^{2} c^{2} d^{2} - 9 \, a b c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (b^{2} c^{3} d^{4} - a b c^{2} d^{5} + {\left (b^{2} c d^{6} - a b d^{7}\right )} x^{2} + 2 \, {\left (b^{2} c^{2} d^{5} - a 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 = 1.71, size = 283, normalized size = 1.72 \begin {gather*} \frac {{\left ({\left (b x + a\right )} {\left (\frac {3 \, {\left (b^{6} c d^{4} - a b^{5} d^{5}\right )} {\left (b x + a\right )}}{b^{4} c d^{5} {\left | b \right |} - a b^{3} d^{6} {\left | b \right |}} + \frac {2 \, {\left (10 \, b^{7} c^{2} d^{3} - 12 \, a b^{6} c d^{4} + 3 \, a^{2} b^{5} d^{5}\right )}}{b^{4} c d^{5} {\left | b \right |} - a b^{3} d^{6} {\left | b \right |}}\right )} + \frac {3 \, {\left (5 \, b^{8} c^{3} d^{2} - 11 \, a b^{7} c^{2} d^{3} + 7 \, a^{2} b^{6} c d^{4} - a^{3} b^{5} d^{5}\right )}}{b^{4} c d^{5} {\left | b \right |} - a b^{3} d^{6} {\left | b \right |}}\right )} \sqrt {b x + a}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} + \frac {{\left (5 \, b^{2} c - a b 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^{3} {\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 = 659, normalized size = 3.99 \begin {gather*} \frac {\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 )-18 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 )+15 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 )-36 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 )+30 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 )-18 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 )+15 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 )+6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,d^{3} x^{2}-6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b c \,d^{2} x^{2}+36 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a c \,d^{2} x -40 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b \,c^{2} d x +26 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,c^{2} d -30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b \,c^{3}\right ) \sqrt {b x +a}}{6 \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^{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 )}^{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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