Optimal. Leaf size=214 \[ \frac {5 (b c-7 a d) (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{9/2} \sqrt {d}}+\frac {5 \sqrt {a+b x} \sqrt {c+d x} (b c-7 a d) (b c-a d)}{8 b^4}+\frac {5 \sqrt {a+b x} (c+d x)^{3/2} (b c-7 a d)}{12 b^3}+\frac {\sqrt {a+b x} (c+d x)^{5/2} (b c-7 a d)}{3 b^2 (b c-a d)}+\frac {2 a (c+d x)^{7/2}}{b \sqrt {a+b x} (b c-a d)} \]
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Rubi [A] time = 0.12, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {78, 50, 63, 217, 206} \begin {gather*} \frac {\sqrt {a+b x} (c+d x)^{5/2} (b c-7 a d)}{3 b^2 (b c-a d)}+\frac {5 \sqrt {a+b x} (c+d x)^{3/2} (b c-7 a d)}{12 b^3}+\frac {5 \sqrt {a+b x} \sqrt {c+d x} (b c-7 a d) (b c-a d)}{8 b^4}+\frac {5 (b c-7 a d) (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{9/2} \sqrt {d}}+\frac {2 a (c+d x)^{7/2}}{b \sqrt {a+b x} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {x (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx &=\frac {2 a (c+d x)^{7/2}}{b (b c-a d) \sqrt {a+b x}}+\frac {(b c-7 a d) \int \frac {(c+d x)^{5/2}}{\sqrt {a+b x}} \, dx}{b (b c-a d)}\\ &=\frac {(b c-7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{3 b^2 (b c-a d)}+\frac {2 a (c+d x)^{7/2}}{b (b c-a d) \sqrt {a+b x}}+\frac {(5 (b c-7 a d)) \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}} \, dx}{6 b^2}\\ &=\frac {5 (b c-7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^3}+\frac {(b c-7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{3 b^2 (b c-a d)}+\frac {2 a (c+d x)^{7/2}}{b (b c-a d) \sqrt {a+b x}}+\frac {(5 (b c-7 a d) (b c-a d)) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx}{8 b^3}\\ &=\frac {5 (b c-7 a d) (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{8 b^4}+\frac {5 (b c-7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^3}+\frac {(b c-7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{3 b^2 (b c-a d)}+\frac {2 a (c+d x)^{7/2}}{b (b c-a d) \sqrt {a+b x}}+\frac {\left (5 (b c-7 a d) (b c-a d)^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 b^4}\\ &=\frac {5 (b c-7 a d) (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{8 b^4}+\frac {5 (b c-7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^3}+\frac {(b c-7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{3 b^2 (b c-a d)}+\frac {2 a (c+d x)^{7/2}}{b (b c-a d) \sqrt {a+b x}}+\frac {\left (5 (b c-7 a d) (b c-a 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 )}{8 b^5}\\ &=\frac {5 (b c-7 a d) (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{8 b^4}+\frac {5 (b c-7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^3}+\frac {(b c-7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{3 b^2 (b c-a d)}+\frac {2 a (c+d x)^{7/2}}{b (b c-a d) \sqrt {a+b x}}+\frac {\left (5 (b c-7 a d) (b c-a 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 )}{8 b^5}\\ &=\frac {5 (b c-7 a d) (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{8 b^4}+\frac {5 (b c-7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^3}+\frac {(b c-7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{3 b^2 (b c-a d)}+\frac {2 a (c+d x)^{7/2}}{b (b c-a d) \sqrt {a+b x}}+\frac {5 (b c-7 a d) (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{9/2} \sqrt {d}}\\ \end {align*}
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Mathematica [A] time = 0.65, size = 175, normalized size = 0.82 \begin {gather*} \frac {\sqrt {c+d x} \left (\frac {105 a^3 d^2+5 a^2 b d (7 d x-38 c)+a b^2 \left (81 c^2-68 c d x-14 d^2 x^2\right )+b^3 x \left (33 c^2+26 c d x+8 d^2 x^2\right )}{\sqrt {a+b x}}+\frac {15 (b c-7 a d) (b c-a d)^{3/2} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{\sqrt {d} \sqrt {\frac {b (c+d x)}{b c-a d}}}\right )}{24 b^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.53, size = 231, normalized size = 1.08 \begin {gather*} \frac {5 (b c-7 a d) (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{8 b^{9/2} \sqrt {d}}+\frac {\sqrt {c+d x} (b c-a d)^2 \left (\frac {48 a b^3 (c+d x)^3}{(a+b x)^3}+\frac {33 b^3 c (c+d x)^2}{(a+b x)^2}-\frac {231 a b^2 d (c+d x)^2}{(a+b x)^2}-\frac {40 b^2 c d (c+d x)}{a+b x}+\frac {280 a b d^2 (c+d x)}{a+b x}-105 a d^3+15 b c d^2\right )}{24 b^4 \sqrt {a+b x} \left (\frac {b (c+d x)}{a+b x}-d\right )^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.47, size = 594, normalized size = 2.78 \begin {gather*} \left [-\frac {15 \, {\left (a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 15 \, a^{3} b c d^{2} - 7 \, a^{4} d^{3} + {\left (b^{4} c^{3} - 9 \, a b^{3} c^{2} d + 15 \, a^{2} b^{2} c d^{2} - 7 \, a^{3} b 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 (8 \, b^{4} d^{3} x^{3} + 81 \, a b^{3} c^{2} d - 190 \, a^{2} b^{2} c d^{2} + 105 \, a^{3} b d^{3} + 2 \, {\left (13 \, b^{4} c d^{2} - 7 \, a b^{3} d^{3}\right )} x^{2} + {\left (33 \, b^{4} c^{2} d - 68 \, a b^{3} c d^{2} + 35 \, a^{2} b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, {\left (b^{6} d x + a b^{5} d\right )}}, -\frac {15 \, {\left (a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 15 \, a^{3} b c d^{2} - 7 \, a^{4} d^{3} + {\left (b^{4} c^{3} - 9 \, a b^{3} c^{2} d + 15 \, a^{2} b^{2} c d^{2} - 7 \, a^{3} b 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 (8 \, b^{4} d^{3} x^{3} + 81 \, a b^{3} c^{2} d - 190 \, a^{2} b^{2} c d^{2} + 105 \, a^{3} b d^{3} + 2 \, {\left (13 \, b^{4} c d^{2} - 7 \, a b^{3} d^{3}\right )} x^{2} + {\left (33 \, b^{4} c^{2} d - 68 \, a b^{3} c d^{2} + 35 \, a^{2} b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (b^{6} d x + a b^{5} d\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.09, size = 370, normalized size = 1.73 \begin {gather*} \frac {1}{24} \, \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 )} d^{2} {\left | b \right |}}{b^{6}} + \frac {13 \, b^{18} c d^{5} {\left | b \right |} - 19 \, a b^{17} d^{6} {\left | b \right |}}{b^{23} d^{4}}\right )} + \frac {3 \, {\left (11 \, b^{19} c^{2} d^{4} {\left | b \right |} - 40 \, a b^{18} c d^{5} {\left | b \right |} + 29 \, a^{2} b^{17} d^{6} {\left | b \right |}\right )}}{b^{23} d^{4}}\right )} + \frac {4 \, {\left (\sqrt {b d} a b^{3} c^{3} {\left | b \right |} - 3 \, \sqrt {b d} a^{2} b^{2} c^{2} d {\left | b \right |} + 3 \, \sqrt {b d} a^{3} b c d^{2} {\left | b \right |} - \sqrt {b d} a^{4} d^{3} {\left | b \right |}\right )}}{{\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 )} b^{5}} - \frac {5 \, {\left (\sqrt {b d} b^{3} c^{3} {\left | b \right |} - 9 \, \sqrt {b d} a b^{2} c^{2} d {\left | b \right |} + 15 \, \sqrt {b d} a^{2} b c d^{2} {\left | b \right |} - 7 \, \sqrt {b d} a^{3} d^{3} {\left | b \right |}\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 )}^{2}\right )}{16 \, b^{6} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 689, normalized size = 3.22 \begin {gather*} -\frac {\sqrt {d x +c}\, \left (105 a^{3} b \,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 )-225 a^{2} b^{2} 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 )+135 a \,b^{3} 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 )-15 b^{4} c^{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 )+105 a^{4} 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 )-225 a^{3} b 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 )+135 a^{2} b^{2} 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 a \,b^{3} 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 )-16 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{3} d^{2} x^{3}+28 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{2} d^{2} x^{2}-52 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{3} c d \,x^{2}-70 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b \,d^{2} x +136 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{2} c d x -66 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{3} c^{2} x -210 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{3} d^{2}+380 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b c d -162 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{2} c^{2}\right )}{48 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {b x +a}\, b^{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\,{\left (c+d\,x\right )}^{5/2}}{{\left (a+b\,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 \left (c + d x\right )^{\frac {5}{2}}}{\left (a + b x\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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