Optimal. Leaf size=267 \[ \frac {2 c^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (4 b c-3 a d) (a+b x)^{5/2}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{4 d^4 (b c-a d)}+\frac {\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{6 d^3 (b c-a d)^2}+\frac {\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 \sqrt {b} d^{9/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.19, antiderivative size = 267, 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 = {91, 79, 52, 65,
223, 212} \begin {gather*} \frac {\left (3 a^2 d^2-30 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 )}{4 \sqrt {b} d^{9/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right )}{4 d^4 (b c-a d)}+\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right )}{6 d^3 (b c-a d)^2}+\frac {2 c^2 (a+b x)^{5/2}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac {4 c (a+b x)^{5/2} (4 b c-3 a d)}{3 d^2 \sqrt {c+d x} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 52
Rule 65
Rule 79
Rule 91
Rule 212
Rule 223
Rubi steps
\begin {align*} \int \frac {x^2 (a+b x)^{3/2}}{(c+d x)^{5/2}} \, dx &=\frac {2 c^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {2 \int \frac {(a+b x)^{3/2} \left (\frac {1}{2} c (5 b c-3 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)^{5/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (4 b c-3 a d) (a+b x)^{5/2}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{3 d^2 (b c-a d)^2}\\ &=\frac {2 c^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (4 b c-3 a d) (a+b x)^{5/2}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{6 d^3 (b c-a d)^2}-\frac {\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{4 d^3 (b c-a d)}\\ &=\frac {2 c^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (4 b c-3 a d) (a+b x)^{5/2}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{4 d^4 (b c-a d)}+\frac {\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{6 d^3 (b c-a d)^2}+\frac {\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 d^4}\\ &=\frac {2 c^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (4 b c-3 a d) (a+b x)^{5/2}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{4 d^4 (b c-a d)}+\frac {\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{6 d^3 (b c-a d)^2}+\frac {\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \text {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 d^4}\\ &=\frac {2 c^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (4 b c-3 a d) (a+b x)^{5/2}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{4 d^4 (b c-a d)}+\frac {\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{6 d^3 (b c-a d)^2}+\frac {\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \text {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 d^4}\\ &=\frac {2 c^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (4 b c-3 a d) (a+b x)^{5/2}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{4 d^4 (b c-a d)}+\frac {\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{6 d^3 (b c-a d)^2}+\frac {\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 \sqrt {b} d^{9/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.30, size = 149, normalized size = 0.56 \begin {gather*} \frac {\sqrt {a+b x} \left (a d \left (55 c^2+78 c d x+15 d^2 x^2\right )-b \left (105 c^3+140 c^2 d x+21 c d^2 x^2-6 d^3 x^3\right )\right )}{12 d^4 (c+d x)^{3/2}}+\frac {\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{4 \sqrt {b} d^{9/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(675\) vs.
\(2(229)=458\).
time = 0.07, size = 676, normalized size = 2.53
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \left (9 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} d^{4} x^{2}-90 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b c \,d^{3} x^{2}+105 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c^{2} d^{2} x^{2}+12 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}\, b \,d^{3} x^{3}+18 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} c \,d^{3} x -180 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b \,c^{2} d^{2} x +210 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c^{3} d x +30 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}\, a \,d^{3} x^{2}-42 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}\, b c \,d^{2} x^{2}+9 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} c^{2} d^{2}-90 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b \,c^{3} d +105 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c^{4}+156 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}\, a c \,d^{2} x -280 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}\, b \,c^{2} d x +110 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}\, a \,c^{2} d -210 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}\, b \,c^{3}\right )}{24 \sqrt {b d}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, d^{4} \left (d x +c \right )^{\frac {3}{2}}}\) | \(676\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.50, size = 594, normalized size = 2.22 \begin {gather*} \left [\frac {3 \, {\left (35 \, b^{2} c^{4} - 30 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} + {\left (35 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{2} + 2 \, {\left (35 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 3 \, 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 (6 \, b^{2} d^{4} x^{3} - 105 \, b^{2} c^{3} d + 55 \, a b c^{2} d^{2} - 3 \, {\left (7 \, b^{2} c d^{3} - 5 \, a b d^{4}\right )} x^{2} - 2 \, {\left (70 \, b^{2} c^{2} d^{2} - 39 \, a b c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (b d^{7} x^{2} + 2 \, b c d^{6} x + b c^{2} d^{5}\right )}}, -\frac {3 \, {\left (35 \, b^{2} c^{4} - 30 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} + {\left (35 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{2} + 2 \, {\left (35 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 3 \, 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 (6 \, b^{2} d^{4} x^{3} - 105 \, b^{2} c^{3} d + 55 \, a b c^{2} d^{2} - 3 \, {\left (7 \, b^{2} c d^{3} - 5 \, a b d^{4}\right )} x^{2} - 2 \, {\left (70 \, b^{2} c^{2} d^{2} - 39 \, a b c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{24 \, {\left (b d^{7} x^{2} + 2 \, b c d^{6} x + b c^{2} d^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.11, size = 393, normalized size = 1.47 \begin {gather*} \frac {{\left ({\left (3 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (b^{6} c d^{6} - a b^{5} d^{7}\right )} {\left (b x + a\right )}}{b^{4} c d^{7} {\left | b \right |} - a b^{3} d^{8} {\left | b \right |}} - \frac {7 \, b^{7} c^{2} d^{5} - 6 \, a b^{6} c d^{6} - a^{2} b^{5} d^{7}}{b^{4} c d^{7} {\left | b \right |} - a b^{3} d^{8} {\left | b \right |}}\right )} - \frac {4 \, {\left (35 \, b^{8} c^{3} d^{4} - 65 \, a b^{7} c^{2} d^{5} + 33 \, a^{2} b^{6} c d^{6} - 3 \, a^{3} b^{5} d^{7}\right )}}{b^{4} c d^{7} {\left | b \right |} - a b^{3} d^{8} {\left | b \right |}}\right )} {\left (b x + a\right )} - \frac {3 \, {\left (35 \, b^{9} c^{4} d^{3} - 100 \, a b^{8} c^{3} d^{4} + 98 \, a^{2} b^{7} c^{2} d^{5} - 36 \, a^{3} b^{6} c d^{6} + 3 \, a^{4} b^{5} d^{7}\right )}}{b^{4} c d^{7} {\left | b \right |} - a b^{3} d^{8} {\left | b \right |}}\right )} \sqrt {b x + a}}{12 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {{\left (35 \, b^{3} c^{2} - 30 \, a b^{2} c d + 3 \, a^{2} b d^{2}\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 )}{4 \, \sqrt {b d} d^{4} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________