Optimal. Leaf size=254 \[ \frac {(b c-a d)^2 \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{9/2} d^{5/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-a d) \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right )}{64 b^4 d^2}+\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right )}{96 b^3 d^2}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (7 a d+3 b c)}{24 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{5/2}}{4 b d} \]
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Rubi [A] time = 0.23, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {90, 80, 50, 63, 217, 206} \begin {gather*} \frac {\sqrt {a+b x} (c+d x)^{3/2} \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right )}{96 b^3 d^2}+\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-a d) \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right )}{64 b^4 d^2}+\frac {(b c-a d)^2 \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{9/2} d^{5/2}}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (7 a d+3 b c)}{24 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{5/2}}{4 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 90
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {x^2 (c+d x)^{3/2}}{\sqrt {a+b x}} \, dx &=\frac {x \sqrt {a+b x} (c+d x)^{5/2}}{4 b d}+\frac {\int \frac {(c+d x)^{3/2} \left (-a c-\frac {1}{2} (3 b c+7 a d) x\right )}{\sqrt {a+b x}} \, dx}{4 b d}\\ &=-\frac {(3 b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{5/2}}{4 b d}+\frac {\left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}} \, dx}{48 b^2 d^2}\\ &=\frac {\left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 b^3 d^2}-\frac {(3 b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{5/2}}{4 b d}+\frac {\left ((b c-a d) \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx}{64 b^3 d^2}\\ &=\frac {(b c-a d) \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^4 d^2}+\frac {\left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 b^3 d^2}-\frac {(3 b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{5/2}}{4 b d}+\frac {\left ((b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 b^4 d^2}\\ &=\frac {(b c-a d) \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^4 d^2}+\frac {\left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 b^3 d^2}-\frac {(3 b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{5/2}}{4 b d}+\frac {\left ((b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right )\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 )}{64 b^5 d^2}\\ &=\frac {(b c-a d) \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^4 d^2}+\frac {\left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 b^3 d^2}-\frac {(3 b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{5/2}}{4 b d}+\frac {\left ((b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right )\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 )}{64 b^5 d^2}\\ &=\frac {(b c-a d) \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^4 d^2}+\frac {\left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 b^3 d^2}-\frac {(3 b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{5/2}}{4 b d}+\frac {(b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{9/2} d^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.69, size = 209, normalized size = 0.82 \begin {gather*} \frac {\sqrt {c+d x} \left (\frac {3 \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) (b c-a d)^{3/2} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{\sqrt {\frac {b (c+d x)}{b c-a d}}}+\sqrt {d} \sqrt {a+b x} \left (-105 a^3 d^3+5 a^2 b d^2 (29 c+14 d x)-a b^2 d \left (15 c^2+92 c d x+56 d^2 x^2\right )+b^3 \left (-9 c^3+6 c^2 d x+72 c d^2 x^2+48 d^3 x^3\right )\right )\right )}{192 b^4 d^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.55, size = 370, normalized size = 1.46 \begin {gather*} \frac {(b c-a d)^2 \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{9/2} d^{5/2}}-\frac {\sqrt {a+b x} (b c-a d)^2 \left (-279 a^2 b^3 d^2+\frac {511 a^2 b^2 d^3 (a+b x)}{c+d x}+\frac {105 a^2 d^5 (a+b x)^3}{(c+d x)^3}-\frac {385 a^2 b d^4 (a+b x)^2}{(c+d x)^2}-\frac {33 b^4 c^2 d (a+b x)}{c+d x}+30 a b^4 c d-\frac {33 b^3 c^2 d^2 (a+b x)^2}{(c+d x)^2}+\frac {146 a b^3 c d^2 (a+b x)}{c+d x}+\frac {9 b^2 c^2 d^3 (a+b x)^3}{(c+d x)^3}-\frac {110 a b^2 c d^3 (a+b x)^2}{(c+d x)^2}+\frac {30 a b c d^4 (a+b x)^3}{(c+d x)^3}+9 b^5 c^2\right )}{192 b^4 d^2 \sqrt {c+d x} \left (b-\frac {d (a+b x)}{c+d x}\right )^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.42, size = 546, normalized size = 2.15 \begin {gather*} \left [\frac {3 \, {\left (3 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 60 \, a^{3} b c d^{3} + 35 \, a^{4} d^{4}\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 (48 \, b^{4} d^{4} x^{3} - 9 \, b^{4} c^{3} d - 15 \, a b^{3} c^{2} d^{2} + 145 \, a^{2} b^{2} c d^{3} - 105 \, a^{3} b d^{4} + 8 \, {\left (9 \, b^{4} c d^{3} - 7 \, a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (3 \, b^{4} c^{2} d^{2} - 46 \, a b^{3} c d^{3} + 35 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, b^{5} d^{3}}, -\frac {3 \, {\left (3 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 60 \, a^{3} b c d^{3} + 35 \, a^{4} d^{4}\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 (48 \, b^{4} d^{4} x^{3} - 9 \, b^{4} c^{3} d - 15 \, a b^{3} c^{2} d^{2} + 145 \, a^{2} b^{2} c d^{3} - 105 \, a^{3} b d^{4} + 8 \, {\left (9 \, b^{4} c d^{3} - 7 \, a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (3 \, b^{4} c^{2} d^{2} - 46 \, a b^{3} c d^{3} + 35 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, b^{5} d^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.28, size = 501, normalized size = 1.97 \begin {gather*} \frac {\frac {8 \, {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{b^{2}} + \frac {b^{6} c d^{3} - 13 \, a b^{5} d^{4}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{7} c^{2} d^{2} + 2 \, a b^{6} c d^{3} - 11 \, a^{2} b^{5} d^{4}\right )}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\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} b d^{2}}\right )} c {\left | b \right |}}{b^{2}} + \frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )}}{b^{3}} + \frac {b^{12} c d^{5} - 25 \, a b^{11} d^{6}}{b^{14} d^{6}}\right )} - \frac {5 \, b^{13} c^{2} d^{4} + 14 \, a b^{12} c d^{5} - 163 \, a^{2} b^{11} d^{6}}{b^{14} d^{6}}\right )} + \frac {3 \, {\left (5 \, b^{14} c^{3} d^{3} + 9 \, a b^{13} c^{2} d^{4} + 15 \, a^{2} b^{12} c d^{5} - 93 \, a^{3} b^{11} d^{6}\right )}}{b^{14} d^{6}}\right )} \sqrt {b x + a} + \frac {3 \, {\left (5 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 35 \, a^{4} d^{4}\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} b^{2} d^{3}}\right )} d {\left | b \right |}}{b^{2}}}{192 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 574, normalized size = 2.26 \begin {gather*} \frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (105 a^{4} d^{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 )-180 a^{3} b c \,d^{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 )+54 a^{2} b^{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 )+12 a \,b^{3} 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 )+9 b^{4} 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 )+96 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{3} d^{3} x^{3}-112 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{2} d^{3} x^{2}+144 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{3} c \,d^{2} x^{2}+140 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b \,d^{3} x -184 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{2} c \,d^{2} x +12 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{3} c^{2} d x -210 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{3} d^{3}+290 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b c \,d^{2}-30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{2} c^{2} d -18 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{3} c^{3}\right )}{384 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{4} 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 = -0.00 \begin {gather*} \int \frac {x^2\,{\left (c+d\,x\right )}^{3/2}}{\sqrt {a+b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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