Optimal. Leaf size=95 \[ \frac {35 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2}}+\frac {35 b}{4 a^4 \sqrt {x}}-\frac {35}{12 a^3 x^{3/2}}+\frac {7}{4 a^2 x^{3/2} (a+b x)}+\frac {1}{2 a x^{3/2} (a+b x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {51, 63, 205} \[ \frac {35 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2}}+\frac {7}{4 a^2 x^{3/2} (a+b x)}+\frac {35 b}{4 a^4 \sqrt {x}}-\frac {35}{12 a^3 x^{3/2}}+\frac {1}{2 a x^{3/2} (a+b x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 205
Rubi steps
\begin {align*} \int \frac {1}{x^{5/2} (a+b x)^3} \, dx &=\frac {1}{2 a x^{3/2} (a+b x)^2}+\frac {7 \int \frac {1}{x^{5/2} (a+b x)^2} \, dx}{4 a}\\ &=\frac {1}{2 a x^{3/2} (a+b x)^2}+\frac {7}{4 a^2 x^{3/2} (a+b x)}+\frac {35 \int \frac {1}{x^{5/2} (a+b x)} \, dx}{8 a^2}\\ &=-\frac {35}{12 a^3 x^{3/2}}+\frac {1}{2 a x^{3/2} (a+b x)^2}+\frac {7}{4 a^2 x^{3/2} (a+b x)}-\frac {(35 b) \int \frac {1}{x^{3/2} (a+b x)} \, dx}{8 a^3}\\ &=-\frac {35}{12 a^3 x^{3/2}}+\frac {35 b}{4 a^4 \sqrt {x}}+\frac {1}{2 a x^{3/2} (a+b x)^2}+\frac {7}{4 a^2 x^{3/2} (a+b x)}+\frac {\left (35 b^2\right ) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{8 a^4}\\ &=-\frac {35}{12 a^3 x^{3/2}}+\frac {35 b}{4 a^4 \sqrt {x}}+\frac {1}{2 a x^{3/2} (a+b x)^2}+\frac {7}{4 a^2 x^{3/2} (a+b x)}+\frac {\left (35 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{4 a^4}\\ &=-\frac {35}{12 a^3 x^{3/2}}+\frac {35 b}{4 a^4 \sqrt {x}}+\frac {1}{2 a x^{3/2} (a+b x)^2}+\frac {7}{4 a^2 x^{3/2} (a+b x)}+\frac {35 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 27, normalized size = 0.28 \[ -\frac {2 \, _2F_1\left (-\frac {3}{2},3;-\frac {1}{2};-\frac {b x}{a}\right )}{3 a^3 x^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 250, normalized size = 2.63 \[ \left [\frac {105 \, {\left (b^{3} x^{4} + 2 \, a b^{2} x^{3} + a^{2} b x^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (105 \, b^{3} x^{3} + 175 \, a b^{2} x^{2} + 56 \, a^{2} b x - 8 \, a^{3}\right )} \sqrt {x}}{24 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}, -\frac {105 \, {\left (b^{3} x^{4} + 2 \, a b^{2} x^{3} + a^{2} b x^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (105 \, b^{3} x^{3} + 175 \, a b^{2} x^{2} + 56 \, a^{2} b x - 8 \, a^{3}\right )} \sqrt {x}}{12 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.10, size = 71, normalized size = 0.75 \[ \frac {35 \, b^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{4}} + \frac {2 \, {\left (9 \, b x - a\right )}}{3 \, a^{4} x^{\frac {3}{2}}} + \frac {11 \, b^{3} x^{\frac {3}{2}} + 13 \, a b^{2} \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 79, normalized size = 0.83 \[ \frac {11 b^{3} x^{\frac {3}{2}}}{4 \left (b x +a \right )^{2} a^{4}}+\frac {13 b^{2} \sqrt {x}}{4 \left (b x +a \right )^{2} a^{3}}+\frac {35 b^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}\, a^{4}}+\frac {6 b}{a^{4} \sqrt {x}}-\frac {2}{3 a^{3} x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.98, size = 86, normalized size = 0.91 \[ \frac {105 \, b^{3} x^{3} + 175 \, a b^{2} x^{2} + 56 \, a^{2} b x - 8 \, a^{3}}{12 \, {\left (a^{4} b^{2} x^{\frac {7}{2}} + 2 \, a^{5} b x^{\frac {5}{2}} + a^{6} x^{\frac {3}{2}}\right )}} + \frac {35 \, b^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.16, size = 80, normalized size = 0.84 \[ \frac {\frac {175\,b^2\,x^2}{12\,a^3}-\frac {2}{3\,a}+\frac {35\,b^3\,x^3}{4\,a^4}+\frac {14\,b\,x}{3\,a^2}}{a^2\,x^{3/2}+b^2\,x^{7/2}+2\,a\,b\,x^{5/2}}+\frac {35\,b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,a^{9/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 138.08, size = 962, normalized size = 10.13 \[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {9}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{3 a^{3} x^{\frac {3}{2}}} & \text {for}\: b = 0 \\- \frac {2}{9 b^{3} x^{\frac {9}{2}}} & \text {for}\: a = 0 \\- \frac {16 i a^{\frac {7}{2}} \sqrt {\frac {1}{b}}}{24 i a^{\frac {13}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 48 i a^{\frac {11}{2}} b x^{\frac {5}{2}} \sqrt {\frac {1}{b}} + 24 i a^{\frac {9}{2}} b^{2} x^{\frac {7}{2}} \sqrt {\frac {1}{b}}} + \frac {112 i a^{\frac {5}{2}} b x \sqrt {\frac {1}{b}}}{24 i a^{\frac {13}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 48 i a^{\frac {11}{2}} b x^{\frac {5}{2}} \sqrt {\frac {1}{b}} + 24 i a^{\frac {9}{2}} b^{2} x^{\frac {7}{2}} \sqrt {\frac {1}{b}}} + \frac {350 i a^{\frac {3}{2}} b^{2} x^{2} \sqrt {\frac {1}{b}}}{24 i a^{\frac {13}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 48 i a^{\frac {11}{2}} b x^{\frac {5}{2}} \sqrt {\frac {1}{b}} + 24 i a^{\frac {9}{2}} b^{2} x^{\frac {7}{2}} \sqrt {\frac {1}{b}}} + \frac {210 i \sqrt {a} b^{3} x^{3} \sqrt {\frac {1}{b}}}{24 i a^{\frac {13}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 48 i a^{\frac {11}{2}} b x^{\frac {5}{2}} \sqrt {\frac {1}{b}} + 24 i a^{\frac {9}{2}} b^{2} x^{\frac {7}{2}} \sqrt {\frac {1}{b}}} + \frac {105 a^{2} b x^{\frac {3}{2}} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{24 i a^{\frac {13}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 48 i a^{\frac {11}{2}} b x^{\frac {5}{2}} \sqrt {\frac {1}{b}} + 24 i a^{\frac {9}{2}} b^{2} x^{\frac {7}{2}} \sqrt {\frac {1}{b}}} - \frac {105 a^{2} b x^{\frac {3}{2}} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{24 i a^{\frac {13}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 48 i a^{\frac {11}{2}} b x^{\frac {5}{2}} \sqrt {\frac {1}{b}} + 24 i a^{\frac {9}{2}} b^{2} x^{\frac {7}{2}} \sqrt {\frac {1}{b}}} + \frac {210 a b^{2} x^{\frac {5}{2}} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{24 i a^{\frac {13}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 48 i a^{\frac {11}{2}} b x^{\frac {5}{2}} \sqrt {\frac {1}{b}} + 24 i a^{\frac {9}{2}} b^{2} x^{\frac {7}{2}} \sqrt {\frac {1}{b}}} - \frac {210 a b^{2} x^{\frac {5}{2}} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{24 i a^{\frac {13}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 48 i a^{\frac {11}{2}} b x^{\frac {5}{2}} \sqrt {\frac {1}{b}} + 24 i a^{\frac {9}{2}} b^{2} x^{\frac {7}{2}} \sqrt {\frac {1}{b}}} + \frac {105 b^{3} x^{\frac {7}{2}} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{24 i a^{\frac {13}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 48 i a^{\frac {11}{2}} b x^{\frac {5}{2}} \sqrt {\frac {1}{b}} + 24 i a^{\frac {9}{2}} b^{2} x^{\frac {7}{2}} \sqrt {\frac {1}{b}}} - \frac {105 b^{3} x^{\frac {7}{2}} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{24 i a^{\frac {13}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 48 i a^{\frac {11}{2}} b x^{\frac {5}{2}} \sqrt {\frac {1}{b}} + 24 i a^{\frac {9}{2}} b^{2} x^{\frac {7}{2}} \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________