Optimal. Leaf size=248 \[ \frac {\sqrt {a+b x} \sqrt {c+d x} \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right )}{8 b d^4}-\frac {(b c-a d) \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{3/2} d^{9/2}}+\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (\frac {a^2 d}{b}+10 a c-\frac {35 b c^2}{d}\right )}{12 d^2 (b c-a d)}+\frac {2 c^2 (a+b x)^{5/2}}{d^2 \sqrt {c+d x} (b c-a d)}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b d^2} \]
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Rubi [A] time = 0.25, antiderivative size = 248, 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 = {89, 80, 50, 63, 217, 206} \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right )}{8 b d^4}-\frac {(b c-a d) \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{3/2} d^{9/2}}+\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (\frac {a^2 d}{b}+10 a c-\frac {35 b c^2}{d}\right )}{12 d^2 (b c-a d)}+\frac {2 c^2 (a+b x)^{5/2}}{d^2 \sqrt {c+d x} (b c-a d)}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 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 (a+b x)^{3/2}}{(c+d x)^{3/2}} \, dx &=\frac {2 c^2 (a+b x)^{5/2}}{d^2 (b c-a d) \sqrt {c+d x}}-\frac {2 \int \frac {(a+b x)^{3/2} \left (\frac {1}{2} c (5 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)^{5/2}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b d^2}-\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{6 b d^2 (b c-a d)}\\ &=\frac {2 c^2 (a+b x)^{5/2}}{d^2 (b c-a d) \sqrt {c+d x}}-\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{12 b d^3 (b c-a d)}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b d^2}+\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{8 b d^3}\\ &=\frac {2 c^2 (a+b x)^{5/2}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b d^4}-\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{12 b d^3 (b c-a d)}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b d^2}-\frac {\left ((b c-a d) \left (35 b^2 c^2-10 a b c d-a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 b d^4}\\ &=\frac {2 c^2 (a+b x)^{5/2}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b d^4}-\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{12 b d^3 (b c-a d)}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b d^2}-\frac {\left ((b c-a d) \left (35 b^2 c^2-10 a b c d-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 )}{8 b^2 d^4}\\ &=\frac {2 c^2 (a+b x)^{5/2}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b d^4}-\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{12 b d^3 (b c-a d)}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b d^2}-\frac {\left ((b c-a d) \left (35 b^2 c^2-10 a b c d-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 )}{8 b^2 d^4}\\ &=\frac {2 c^2 (a+b x)^{5/2}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b d^4}-\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{12 b d^3 (b c-a d)}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b d^2}-\frac {(b c-a d) \left (35 b^2 c^2-10 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 )}{8 b^{3/2} d^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.49, size = 233, normalized size = 0.94 \begin {gather*} \frac {\frac {b \sqrt {d} \left (3 a^3 d^2 (c+d x)+a^2 b d \left (-100 c^2-35 c d x+17 d^2 x^2\right )+a b^2 \left (105 c^3-65 c^2 d x-52 c d^2 x^2+22 d^3 x^3\right )+b^3 x \left (105 c^3+35 c^2 d x-14 c d^2 x^2+8 d^3 x^3\right )\right )}{\sqrt {a+b x}}-3 (b c-a d)^{3/2} \left (-a^2 d^2-10 a b c d+35 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 )}{24 b^2 d^{9/2} \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.62, size = 234, normalized size = 0.94 \begin {gather*} \frac {\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}} \left (3 a^2 d^2 (c+d x)-48 a b c^2 d-66 a b c d (c+d x)+14 a b d (c+d x)^2+48 b^2 c^3+87 b^2 c^2 (c+d x)-38 b^2 c (c+d x)^2+8 b^2 (c+d x)^3\right )}{24 b d^4 \sqrt {c+d x}}+\frac {\sqrt {\frac {b}{d}} \left (a^3 d^3+9 a^2 b c d^2-45 a b^2 c^2 d+35 b^3 c^3\right ) \log \left (\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}-\sqrt {\frac {b}{d}} \sqrt {c+d x}\right )}{8 b^2 d^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.86, size = 600, normalized size = 2.42 \begin {gather*} \left [\frac {3 \, {\left (35 \, b^{3} c^{4} - 45 \, a b^{2} c^{3} d + 9 \, a^{2} b c^{2} d^{2} + a^{3} c d^{3} + {\left (35 \, b^{3} c^{3} d - 45 \, a b^{2} c^{2} d^{2} + 9 \, a^{2} b c d^{3} + a^{3} d^{4}\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 (8 \, b^{3} d^{4} x^{3} + 105 \, b^{3} c^{3} d - 100 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - 14 \, {\left (b^{3} c d^{3} - a b^{2} d^{4}\right )} x^{2} + {\left (35 \, b^{3} c^{2} d^{2} - 38 \, a b^{2} c d^{3} + 3 \, a^{2} b d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, {\left (b^{2} d^{6} x + b^{2} c d^{5}\right )}}, \frac {3 \, {\left (35 \, b^{3} c^{4} - 45 \, a b^{2} c^{3} d + 9 \, a^{2} b c^{2} d^{2} + a^{3} c d^{3} + {\left (35 \, b^{3} c^{3} d - 45 \, a b^{2} c^{2} d^{2} + 9 \, a^{2} b c d^{3} + a^{3} d^{4}\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 (8 \, b^{3} d^{4} x^{3} + 105 \, b^{3} c^{3} d - 100 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - 14 \, {\left (b^{3} c d^{3} - a b^{2} d^{4}\right )} x^{2} + {\left (35 \, b^{3} c^{2} d^{2} - 38 \, a b^{2} c d^{3} + 3 \, a^{2} b d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (b^{2} d^{6} x + b^{2} c d^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.62, size = 282, normalized size = 1.14 \begin {gather*} \frac {{\left ({\left (b x + a\right )} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{d {\left | b \right |}} - \frac {7 \, b^{3} c d^{5} + 5 \, a b^{2} d^{6}}{b^{2} d^{7} {\left | b \right |}}\right )} + \frac {35 \, b^{4} c^{2} d^{4} - 10 \, a b^{3} c d^{5} - a^{2} b^{2} d^{6}}{b^{2} d^{7} {\left | b \right |}}\right )} + \frac {3 \, {\left (35 \, b^{5} c^{3} d^{3} - 45 \, a b^{4} c^{2} d^{4} + 9 \, a^{2} b^{3} c d^{5} + a^{3} b^{2} d^{6}\right )}}{b^{2} d^{7} {\left | b \right |}}\right )} \sqrt {b x + a}}{24 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} + \frac {{\left (35 \, b^{3} c^{3} - 45 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + 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 )}{8 \, \sqrt {b d} d^{4} {\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 = 692, normalized size = 2.79 \begin {gather*} -\frac {\sqrt {b x +a}\, \left (3 a^{3} d^{4} 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 )+27 a^{2} b 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 )-135 a \,b^{2} 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 )+105 b^{3} 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^{3} 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 )+27 a^{2} b \,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 )-135 a \,b^{2} 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 )+105 b^{3} 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 )-16 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{2} d^{3} x^{3}-28 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a b \,d^{3} x^{2}+28 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{2} c \,d^{2} x^{2}-6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} d^{3} x +76 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a b c \,d^{2} x -70 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{2} c^{2} d x -6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} c \,d^{2}+200 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a b \,c^{2} d -210 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{2} c^{3}\right )}{48 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {d x +c}\, b \,d^{4}} \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 (a+b\,x\right )}^{3/2}}{{\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} \left (a + b x\right )^{\frac {3}{2}}}{\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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