Optimal. Leaf size=171 \[ \frac {3 (b c-5 a d) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{7/2} \sqrt {d}}+\frac {3 \sqrt {a+b x} \sqrt {c+d x} (b c-5 a d)}{4 b^3}+\frac {\sqrt {a+b x} (c+d x)^{3/2} (b c-5 a d)}{2 b^2 (b c-a d)}+\frac {2 a (c+d x)^{5/2}}{b \sqrt {a+b x} (b c-a d)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ \frac {\sqrt {a+b x} (c+d x)^{3/2} (b c-5 a d)}{2 b^2 (b c-a d)}+\frac {3 \sqrt {a+b x} \sqrt {c+d x} (b c-5 a d)}{4 b^3}+\frac {3 (b c-5 a d) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{7/2} \sqrt {d}}+\frac {2 a (c+d x)^{5/2}}{b \sqrt {a+b x} (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 78
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {x (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx &=\frac {2 a (c+d x)^{5/2}}{b (b c-a d) \sqrt {a+b x}}+\frac {(b c-5 a d) \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}} \, dx}{b (b c-a d)}\\ &=\frac {(b c-5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{2 b^2 (b c-a d)}+\frac {2 a (c+d x)^{5/2}}{b (b c-a d) \sqrt {a+b x}}+\frac {(3 (b c-5 a d)) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx}{4 b^2}\\ &=\frac {3 (b c-5 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^3}+\frac {(b c-5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{2 b^2 (b c-a d)}+\frac {2 a (c+d x)^{5/2}}{b (b c-a d) \sqrt {a+b x}}+\frac {(3 (b c-5 a d) (b c-a d)) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 b^3}\\ &=\frac {3 (b c-5 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^3}+\frac {(b c-5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{2 b^2 (b c-a d)}+\frac {2 a (c+d x)^{5/2}}{b (b c-a d) \sqrt {a+b x}}+\frac {(3 (b c-5 a d) (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 )}{4 b^4}\\ &=\frac {3 (b c-5 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^3}+\frac {(b c-5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{2 b^2 (b c-a d)}+\frac {2 a (c+d x)^{5/2}}{b (b c-a d) \sqrt {a+b x}}+\frac {(3 (b c-5 a d) (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 )}{4 b^4}\\ &=\frac {3 (b c-5 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^3}+\frac {(b c-5 a d) \sqrt {a+b x} (c+d x)^{3/2}}{2 b^2 (b c-a d)}+\frac {2 a (c+d x)^{5/2}}{b (b c-a d) \sqrt {a+b x}}+\frac {3 (b c-5 a d) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{7/2} \sqrt {d}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.43, size = 134, normalized size = 0.78 \[ \frac {\sqrt {c+d x} \left (\frac {-15 a^2 d+a b (13 c-5 d x)+b^2 x (5 c+2 d x)}{\sqrt {a+b x}}+\frac {3 (b c-5 a d) \sqrt {b c-a d} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{\sqrt {d} \sqrt {\frac {b (c+d x)}{b c-a d}}}\right )}{4 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.05, size = 434, normalized size = 2.54 \[ \left [\frac {3 \, {\left (a b^{2} c^{2} - 6 \, a^{2} b c d + 5 \, a^{3} d^{2} + {\left (b^{3} c^{2} - 6 \, a b^{2} c d + 5 \, a^{2} b d^{2}\right )} x\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (2 \, b^{3} d^{2} x^{2} + 13 \, a b^{2} c d - 15 \, a^{2} b d^{2} + 5 \, {\left (b^{3} c d - a b^{2} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, {\left (b^{5} d x + a b^{4} d\right )}}, -\frac {3 \, {\left (a b^{2} c^{2} - 6 \, a^{2} b c d + 5 \, a^{3} d^{2} + {\left (b^{3} c^{2} - 6 \, a b^{2} c d + 5 \, a^{2} b d^{2}\right )} x\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (2 \, b^{3} d^{2} x^{2} + 13 \, a b^{2} c d - 15 \, a^{2} b d^{2} + 5 \, {\left (b^{3} c d - a b^{2} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, {\left (b^{5} d x + a b^{4} d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.73, size = 270, normalized size = 1.58 \[ \frac {1}{4} \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (\frac {2 \, {\left (b x + a\right )} d {\left | b \right |}}{b^{5}} + \frac {5 \, b^{10} c d^{2} {\left | b \right |} - 9 \, a b^{9} d^{3} {\left | b \right |}}{b^{14} d^{2}}\right )} + \frac {4 \, {\left (\sqrt {b d} a b^{2} c^{2} {\left | b \right |} - 2 \, \sqrt {b d} a^{2} b c d {\left | b \right |} + \sqrt {b d} a^{3} d^{2} {\left | b \right |}\right )}}{{\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )} b^{4}} - \frac {3 \, {\left (\sqrt {b d} b^{2} c^{2} {\left | b \right |} - 6 \, \sqrt {b d} a b c d {\left | b \right |} + 5 \, \sqrt {b d} a^{2} d^{2} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{8 \, b^{5} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 455, normalized size = 2.66 \[ \frac {\sqrt {d x +c}\, \left (15 a^{2} b \,d^{2} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-18 a \,b^{2} c d x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+3 b^{3} c^{2} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+15 a^{3} d^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-18 a^{2} b c d \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+3 a \,b^{2} c^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+4 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{2} d \,x^{2}-10 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a b d x +10 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{2} c x -30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} d +26 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a b c \right )}{8 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {b x +a}\, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,{\left (c+d\,x\right )}^{3/2}}{{\left (a+b\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (c + d x\right )^{\frac {3}{2}}}{\left (a + b x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________