Optimal. Leaf size=218 \[ -\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 a^2 d^2-2 b d x (35 b c-31 a d)-100 a b c d+105 b^2 c^2\right )}{12 b d^4 (b c-a d)}+\frac {\left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2} d^{9/2}}-\frac {2 x^2 \sqrt {a+b x} (7 b c-6 a d)}{3 d^2 \sqrt {c+d x} (b c-a d)}-\frac {2 x^3 \sqrt {a+b x}}{3 d (c+d x)^{3/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.19, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {97, 150, 147, 63, 217, 206} \begin {gather*} -\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 a^2 d^2-2 b d x (35 b c-31 a d)-100 a b c d+105 b^2 c^2\right )}{12 b d^4 (b c-a d)}+\frac {\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 )}{4 b^{3/2} d^{9/2}}-\frac {2 x^2 \sqrt {a+b x} (7 b c-6 a d)}{3 d^2 \sqrt {c+d x} (b c-a d)}-\frac {2 x^3 \sqrt {a+b x}}{3 d (c+d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 97
Rule 147
Rule 150
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {x^3 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx &=-\frac {2 x^3 \sqrt {a+b x}}{3 d (c+d x)^{3/2}}+\frac {2 \int \frac {x^2 \left (3 a+\frac {7 b x}{2}\right )}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 d}\\ &=-\frac {2 x^3 \sqrt {a+b x}}{3 d (c+d x)^{3/2}}-\frac {2 (7 b c-6 a d) x^2 \sqrt {a+b x}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {4 \int \frac {x \left (-a (7 b c-6 a d)-\frac {1}{4} b (35 b c-31 a d) x\right )}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 d^2 (b c-a d)}\\ &=-\frac {2 x^3 \sqrt {a+b x}}{3 d (c+d x)^{3/2}}-\frac {2 (7 b c-6 a d) x^2 \sqrt {a+b x}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^2 c^2-100 a b c d+3 a^2 d^2-2 b d (35 b c-31 a d) x\right )}{12 b d^4 (b c-a d)}+\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 b d^4}\\ &=-\frac {2 x^3 \sqrt {a+b x}}{3 d (c+d x)^{3/2}}-\frac {2 (7 b c-6 a d) x^2 \sqrt {a+b x}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^2 c^2-100 a b c d+3 a^2 d^2-2 b d (35 b c-31 a d) x\right )}{12 b d^4 (b c-a d)}+\frac {\left (35 b^2 c^2-10 a b c d-a^2 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 )}{4 b^2 d^4}\\ &=-\frac {2 x^3 \sqrt {a+b x}}{3 d (c+d x)^{3/2}}-\frac {2 (7 b c-6 a d) x^2 \sqrt {a+b x}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^2 c^2-100 a b c d+3 a^2 d^2-2 b d (35 b c-31 a d) x\right )}{12 b d^4 (b c-a d)}+\frac {\left (35 b^2 c^2-10 a b c d-a^2 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 )}{4 b^2 d^4}\\ &=-\frac {2 x^3 \sqrt {a+b x}}{3 d (c+d x)^{3/2}}-\frac {2 (7 b c-6 a d) x^2 \sqrt {a+b x}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^2 c^2-100 a b c d+3 a^2 d^2-2 b d (35 b c-31 a d) x\right )}{12 b d^4 (b c-a d)}+\frac {\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 )}{4 b^{3/2} d^{9/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.30, size = 249, normalized size = 1.14 \begin {gather*} \frac {\frac {3 (c+d x)^2 \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \left (\sqrt {a+b x} (b c-a d) \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )-\sqrt {d} (a+b x) \sqrt {b c-a d} \sqrt {\frac {b (c+d x)}{b c-a d}}\right )}{d^{7/2} (b c-a d)^{3/2} \sqrt {\frac {b (c+d x)}{b c-a d}}}+\frac {2 c (a+b x)^2 (3 a d (c+2 d x)-7 b c (5 c+6 d x))}{d^2 (a d-b c)}+6 x^2 (a+b x)^2}{12 b d \sqrt {a+b x} (c+d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.41, size = 324, normalized size = 1.49 \begin {gather*} \frac {\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 )}{4 b^{3/2} d^{9/2}}-\frac {\sqrt {a+b x} \left (\frac {3 a^3 d^4 (a+b x)}{c+d x}+3 a^3 b d^3+27 a^2 b^2 c d^2-\frac {45 a^2 b c d^3 (a+b x)}{c+d x}-\frac {175 b^3 c^3 d (a+b x)}{c+d x}-135 a b^3 c^2 d+\frac {56 b^2 c^3 d^2 (a+b x)^2}{(c+d x)^2}+\frac {225 a b^2 c^2 d^2 (a+b x)}{c+d x}+\frac {8 b c^3 d^3 (a+b x)^3}{(c+d x)^3}-\frac {72 a b c^2 d^3 (a+b x)^2}{(c+d x)^2}+105 b^4 c^3\right )}{12 b d^4 \sqrt {c+d x} (b c-a d) \left (b-\frac {d (a+b x)}{c+d x}\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 2.50, size = 866, normalized size = 3.97 \begin {gather*} \left [-\frac {3 \, {\left (35 \, b^{3} c^{5} - 45 \, a b^{2} c^{4} d + 9 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3} + {\left (35 \, b^{3} c^{3} d^{2} - 45 \, a b^{2} c^{2} d^{3} + 9 \, a^{2} b c d^{4} + a^{3} d^{5}\right )} x^{2} + 2 \, {\left (35 \, b^{3} c^{4} d - 45 \, a b^{2} c^{3} d^{2} + 9 \, a^{2} b c^{2} d^{3} + a^{3} c 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 (105 \, b^{3} c^{4} d - 100 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - 6 \, {\left (b^{3} c d^{4} - a b^{2} d^{5}\right )} x^{3} + 3 \, {\left (7 \, b^{3} c^{2} d^{3} - 8 \, a b^{2} c d^{4} + a^{2} b d^{5}\right )} x^{2} + 2 \, {\left (70 \, b^{3} c^{3} d^{2} - 69 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (b^{3} c^{3} d^{5} - a b^{2} c^{2} d^{6} + {\left (b^{3} c d^{7} - a b^{2} d^{8}\right )} x^{2} + 2 \, {\left (b^{3} c^{2} d^{6} - a b^{2} c d^{7}\right )} x\right )}}, -\frac {3 \, {\left (35 \, b^{3} c^{5} - 45 \, a b^{2} c^{4} d + 9 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3} + {\left (35 \, b^{3} c^{3} d^{2} - 45 \, a b^{2} c^{2} d^{3} + 9 \, a^{2} b c d^{4} + a^{3} d^{5}\right )} x^{2} + 2 \, {\left (35 \, b^{3} c^{4} d - 45 \, a b^{2} c^{3} d^{2} + 9 \, a^{2} b c^{2} d^{3} + a^{3} c 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 (105 \, b^{3} c^{4} d - 100 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - 6 \, {\left (b^{3} c d^{4} - a b^{2} d^{5}\right )} x^{3} + 3 \, {\left (7 \, b^{3} c^{2} d^{3} - 8 \, a b^{2} c d^{4} + a^{2} b d^{5}\right )} x^{2} + 2 \, {\left (70 \, b^{3} c^{3} d^{2} - 69 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{24 \, {\left (b^{3} c^{3} d^{5} - a b^{2} c^{2} d^{6} + {\left (b^{3} c d^{7} - a b^{2} d^{8}\right )} x^{2} + 2 \, {\left (b^{3} c^{2} d^{6} - a b^{2} c d^{7}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.86, size = 407, normalized size = 1.87 \begin {gather*} \frac {{\left ({\left (3 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (b^{5} c d^{6} {\left | b \right |} - a b^{4} d^{7} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{6} c d^{7} - a b^{5} d^{8}} - \frac {7 \, b^{6} c^{2} d^{5} {\left | b \right |} - 2 \, a b^{5} c d^{6} {\left | b \right |} - 5 \, a^{2} b^{4} d^{7} {\left | b \right |}}{b^{6} c d^{7} - a b^{5} d^{8}}\right )} - \frac {4 \, {\left (35 \, b^{7} c^{3} d^{4} {\left | b \right |} - 45 \, a b^{6} c^{2} d^{5} {\left | b \right |} + 9 \, a^{2} b^{5} c d^{6} {\left | b \right |} + 3 \, a^{3} b^{4} d^{7} {\left | b \right |}\right )}}{b^{6} c d^{7} - a b^{5} d^{8}}\right )} {\left (b x + a\right )} - \frac {3 \, {\left (35 \, b^{8} c^{4} d^{3} {\left | b \right |} - 80 \, a b^{7} c^{3} d^{4} {\left | b \right |} + 54 \, a^{2} b^{6} c^{2} d^{5} {\left | b \right |} - 8 \, a^{3} b^{5} c d^{6} {\left | b \right |} - a^{4} b^{4} d^{7} {\left | b \right |}\right )}}{b^{6} c d^{7} - a b^{5} d^{8}}\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^{2} c^{2} {\left | b \right |} - 10 \, a b c d {\left | b \right |} - 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 |}\right )}{4 \, \sqrt {b d} b^{2} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.03, size = 986, normalized size = 4.52 \begin {gather*} -\frac {\left (3 a^{3} d^{5} x^{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 )+27 a^{2} b c \,d^{4} x^{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^{2} d^{3} x^{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 )+105 b^{3} c^{3} d^{2} x^{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 )+6 a^{3} c \,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 )+54 a^{2} b \,c^{2} 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 )-270 a \,b^{2} c^{3} 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 )+210 b^{3} c^{4} 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^{2} 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^{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 )-135 a \,b^{2} c^{4} 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 )-12 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a b \,d^{4} x^{3}+105 b^{3} c^{5} \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 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{2} c \,d^{3} x^{3}-6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} d^{4} x^{2}+48 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c \,d^{3} x^{2}-42 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2} d^{2} x^{2}-12 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} c \,d^{3} x +276 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{2} d^{2} x -280 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{3} d x -6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} c^{2} d^{2}+200 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{3} d -210 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{4}\right ) \sqrt {b x +a}}{24 \left (a d -b c \right ) \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \left (d x +c \right )^{\frac {3}{2}} b \,d^{4}} \end {gather*}
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]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3\,\sqrt {a+b\,x}}{{\left (c+d\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________