Optimal. Leaf size=116 \[ \frac {2 d^2 \sqrt {a+b x}}{(d+e x)^{3/2} (b d-a e)}+\frac {4 d \sqrt {a+b x} (3 b d-2 a e)}{\sqrt {d+e x} (b d-a e)^2}+\frac {16 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} \sqrt {e}} \]
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Rubi [A] time = 0.12, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {949, 78, 63, 217, 206} \[ \frac {2 d^2 \sqrt {a+b x}}{(d+e x)^{3/2} (b d-a e)}+\frac {4 d \sqrt {a+b x} (3 b d-2 a e)}{\sqrt {d+e x} (b d-a e)^2}+\frac {16 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} \sqrt {e}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 206
Rule 217
Rule 949
Rubi steps
\begin {align*} \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{5/2}} \, dx &=\frac {2 d^2 \sqrt {a+b x}}{(b d-a e) (d+e x)^{3/2}}+\frac {2 \int \frac {3 d (7 b d-6 a e)+12 e (b d-a e) x}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx}{3 (b d-a e)}\\ &=\frac {2 d^2 \sqrt {a+b x}}{(b d-a e) (d+e x)^{3/2}}+\frac {4 d (3 b d-2 a e) \sqrt {a+b x}}{(b d-a e)^2 \sqrt {d+e x}}+8 \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx\\ &=\frac {2 d^2 \sqrt {a+b x}}{(b d-a e) (d+e x)^{3/2}}+\frac {4 d (3 b d-2 a e) \sqrt {a+b x}}{(b d-a e)^2 \sqrt {d+e x}}+\frac {16 \operatorname {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{b}\\ &=\frac {2 d^2 \sqrt {a+b x}}{(b d-a e) (d+e x)^{3/2}}+\frac {4 d (3 b d-2 a e) \sqrt {a+b x}}{(b d-a e)^2 \sqrt {d+e x}}+\frac {16 \operatorname {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{b}\\ &=\frac {2 d^2 \sqrt {a+b x}}{(b d-a e) (d+e x)^{3/2}}+\frac {4 d (3 b d-2 a e) \sqrt {a+b x}}{(b d-a e)^2 \sqrt {d+e x}}+\frac {16 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} \sqrt {e}}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 128, normalized size = 1.10 \[ \frac {2 \left (\frac {8 (b d-a e)^{3/2} \left (\frac {b (d+e x)}{b d-a e}\right )^{3/2} \sinh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b d-a e}}\right )}{b^2 \sqrt {e}}+\frac {d \sqrt {a+b x} (b d (7 d+6 e x)-a e (5 d+4 e x))}{(b d-a e)^2}\right )}{(d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.59, size = 665, normalized size = 5.73 \[ \left [\frac {2 \, {\left (2 \, {\left (b^{2} d^{4} - 2 \, a b d^{3} e + a^{2} d^{2} e^{2} + {\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x^{2} + 2 \, {\left (b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) + {\left (7 \, b^{2} d^{3} e - 5 \, a b d^{2} e^{2} + 2 \, {\left (3 \, b^{2} d^{2} e^{2} - 2 \, a b d e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}\right )}}{b^{3} d^{4} e - 2 \, a b^{2} d^{3} e^{2} + a^{2} b d^{2} e^{3} + {\left (b^{3} d^{2} e^{3} - 2 \, a b^{2} d e^{4} + a^{2} b e^{5}\right )} x^{2} + 2 \, {\left (b^{3} d^{3} e^{2} - 2 \, a b^{2} d^{2} e^{3} + a^{2} b d e^{4}\right )} x}, -\frac {2 \, {\left (4 \, {\left (b^{2} d^{4} - 2 \, a b d^{3} e + a^{2} d^{2} e^{2} + {\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x^{2} + 2 \, {\left (b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) - {\left (7 \, b^{2} d^{3} e - 5 \, a b d^{2} e^{2} + 2 \, {\left (3 \, b^{2} d^{2} e^{2} - 2 \, a b d e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}\right )}}{b^{3} d^{4} e - 2 \, a b^{2} d^{3} e^{2} + a^{2} b d^{2} e^{3} + {\left (b^{3} d^{2} e^{3} - 2 \, a b^{2} d e^{4} + a^{2} b e^{5}\right )} x^{2} + 2 \, {\left (b^{3} d^{3} e^{2} - 2 \, a b^{2} d^{2} e^{3} + a^{2} b d e^{4}\right )} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 601, normalized size = 5.18 \[ \frac {2 \sqrt {b x +a}\, \left (4 a^{2} e^{4} x^{2} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-8 a b d \,e^{3} x^{2} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+4 b^{2} d^{2} e^{2} x^{2} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+8 a^{2} d \,e^{3} x \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-16 a b \,d^{2} e^{2} x \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+8 b^{2} d^{3} e x \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+4 a^{2} d^{2} e^{2} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-8 a b \,d^{3} e \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+4 b^{2} d^{4} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-4 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a d \,e^{2} x +6 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b \,d^{2} e x -5 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a \,d^{2} e +7 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b \,d^{3}\right )}{\sqrt {b e}\, \left (a e -b d \right )^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \left (e x +d \right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {15\,d^2+20\,d\,e\,x+8\,e^2\,x^2}{\sqrt {a+b\,x}\,{\left (d+e\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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