Optimal. Leaf size=101 \[ -\frac {2 (c+d x)^{3/2}}{7 (b c-a d) (a+b x)^{7/2}}+\frac {8 d (c+d x)^{3/2}}{35 (b c-a d)^2 (a+b x)^{5/2}}-\frac {16 d^2 (c+d x)^{3/2}}{105 (b c-a d)^3 (a+b x)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37}
\begin {gather*} -\frac {16 d^2 (c+d x)^{3/2}}{105 (a+b x)^{3/2} (b c-a d)^3}+\frac {8 d (c+d x)^{3/2}}{35 (a+b x)^{5/2} (b c-a d)^2}-\frac {2 (c+d x)^{3/2}}{7 (a+b x)^{7/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x}}{(a+b x)^{9/2}} \, dx &=-\frac {2 (c+d x)^{3/2}}{7 (b c-a d) (a+b x)^{7/2}}-\frac {(4 d) \int \frac {\sqrt {c+d x}}{(a+b x)^{7/2}} \, dx}{7 (b c-a d)}\\ &=-\frac {2 (c+d x)^{3/2}}{7 (b c-a d) (a+b x)^{7/2}}+\frac {8 d (c+d x)^{3/2}}{35 (b c-a d)^2 (a+b x)^{5/2}}+\frac {\left (8 d^2\right ) \int \frac {\sqrt {c+d x}}{(a+b x)^{5/2}} \, dx}{35 (b c-a d)^2}\\ &=-\frac {2 (c+d x)^{3/2}}{7 (b c-a d) (a+b x)^{7/2}}+\frac {8 d (c+d x)^{3/2}}{35 (b c-a d)^2 (a+b x)^{5/2}}-\frac {16 d^2 (c+d x)^{3/2}}{105 (b c-a d)^3 (a+b x)^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.09, size = 77, normalized size = 0.76 \begin {gather*} -\frac {2 (c+d x)^{3/2} \left (35 a^2 d^2+14 a b d (-3 c+2 d x)+b^2 \left (15 c^2-12 c d x+8 d^2 x^2\right )\right )}{105 (b c-a d)^3 (a+b x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(167\) vs.
\(2(83)=166\).
time = 0.16, size = 168, normalized size = 1.66
method | result | size |
gosper | \(\frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (8 b^{2} x^{2} d^{2}+28 a b \,d^{2} x -12 b^{2} c d x +35 a^{2} d^{2}-42 a b c d +15 b^{2} c^{2}\right )}{105 \left (b x +a \right )^{\frac {7}{2}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(105\) |
default | \(-\frac {\sqrt {d x +c}}{3 b \left (b x +a \right )^{\frac {7}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {2 \sqrt {d x +c}}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}}}-\frac {6 d \left (-\frac {2 \sqrt {d x +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\right )}{6 b}\) | \(168\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 337 vs.
\(2 (83) = 166\).
time = 1.51, size = 337, normalized size = 3.34 \begin {gather*} -\frac {2 \, {\left (8 \, b^{2} d^{3} x^{3} + 15 \, b^{2} c^{3} - 42 \, a b c^{2} d + 35 \, a^{2} c d^{2} - 4 \, {\left (b^{2} c d^{2} - 7 \, a b d^{3}\right )} x^{2} + {\left (3 \, b^{2} c^{2} d - 14 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{105 \, {\left (a^{4} b^{3} c^{3} - 3 \, a^{5} b^{2} c^{2} d + 3 \, a^{6} b c d^{2} - a^{7} d^{3} + {\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} x^{4} + 4 \, {\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} x^{3} + 6 \, {\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} x^{2} + 4 \, {\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
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]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 689 vs.
\(2 (83) = 166\).
time = 2.01, size = 689, normalized size = 6.82 \begin {gather*} -\frac {32 \, {\left (\sqrt {b d} b^{10} c^{4} d^{3} - 4 \, \sqrt {b d} a b^{9} c^{3} d^{4} + 6 \, \sqrt {b d} a^{2} b^{8} c^{2} d^{5} - 4 \, \sqrt {b d} a^{3} b^{7} c d^{6} + \sqrt {b d} a^{4} b^{6} d^{7} - 7 \, \sqrt {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} b^{8} c^{3} d^{3} + 21 \, \sqrt {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} a b^{7} c^{2} d^{4} - 21 \, \sqrt {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} a^{2} b^{6} c d^{5} + 7 \, \sqrt {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} a^{3} b^{5} d^{6} + 21 \, \sqrt {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 )}^{4} b^{6} c^{2} d^{3} - 42 \, \sqrt {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 )}^{4} a b^{5} c d^{4} + 21 \, \sqrt {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 )}^{4} a^{2} b^{4} d^{5} + 35 \, \sqrt {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 )}^{6} b^{4} c d^{3} - 35 \, \sqrt {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 )}^{6} a b^{3} d^{4} + 70 \, \sqrt {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 )}^{8} b^{2} d^{3}\right )} {\left | b \right |}}{105 \, {\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 )}^{7} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.97, size = 203, normalized size = 2.01 \begin {gather*} \frac {\sqrt {c+d\,x}\,\left (\frac {70\,a^2\,c\,d^2-84\,a\,b\,c^2\,d+30\,b^2\,c^3}{105\,b^3\,{\left (a\,d-b\,c\right )}^3}+\frac {x\,\left (70\,a^2\,d^3-28\,a\,b\,c\,d^2+6\,b^2\,c^2\,d\right )}{105\,b^3\,{\left (a\,d-b\,c\right )}^3}+\frac {16\,d^3\,x^3}{105\,b\,{\left (a\,d-b\,c\right )}^3}+\frac {8\,d^2\,x^2\,\left (7\,a\,d-b\,c\right )}{105\,b^2\,{\left (a\,d-b\,c\right )}^3}\right )}{x^3\,\sqrt {a+b\,x}+\frac {a^3\,\sqrt {a+b\,x}}{b^3}+\frac {3\,a\,x^2\,\sqrt {a+b\,x}}{b}+\frac {3\,a^2\,x\,\sqrt {a+b\,x}}{b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________