Optimal. Leaf size=150 \[ -\frac {2 (c d f-a e g) (d+e x)^{3/2}}{c d \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {2 \left (2 a e^2 g-c d (e f+d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^2 d^2 \left (c d^2-a e^2\right ) \sqrt {d+e x}} \]
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Rubi [A]
time = 0.10, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {802, 662}
\begin {gather*} -\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (d g+e f)\right )}{c^2 d^2 \sqrt {d+e x} \left (c d^2-a e^2\right )}-\frac {2 (d+e x)^{3/2} (c d f-a e g)}{c d \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 662
Rule 802
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2} (f+g x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=-\frac {2 (c d f-a e g) (d+e x)^{3/2}}{c d \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (2 \left (-\frac {1}{2} e \left (2 c d e f-\left (c d^2+a e^2\right ) g\right )+\frac {3}{2} \left (c d e^2 f+\left (c d^2 e-e \left (c d^2+a e^2\right )\right ) g\right )\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{c d \left (2 c d^2 e-e \left (c d^2+a e^2\right )\right )}\\ &=-\frac {2 (c d f-a e g) (d+e x)^{3/2}}{c d \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {2 \left (2 a e^2 g-c d (e f+d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^2 d^2 \left (c d^2-a e^2\right ) \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 51, normalized size = 0.34 \begin {gather*} \frac {2 \sqrt {d+e x} (2 a e g+c d (-f+g x))}{c^2 d^2 \sqrt {(a e+c d x) (d+e x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 58, normalized size = 0.39
method | result | size |
default | \(\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (c d g x +2 a e g -c d f \right )}{\sqrt {e x +d}\, \left (c d x +a e \right ) c^{2} d^{2}}\) | \(58\) |
gosper | \(\frac {2 \left (c d x +a e \right ) \left (c d g x +2 a e g -c d f \right ) \left (e x +d \right )^{\frac {3}{2}}}{c^{2} d^{2} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 51, normalized size = 0.34 \begin {gather*} -\frac {2 \, f}{\sqrt {c d x + a e} c d} + \frac {2 \, {\left (c d x + 2 \, a e\right )} g}{\sqrt {c d x + a e} c^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.88, size = 98, normalized size = 0.65 \begin {gather*} \frac {2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (c d g x - c d f + 2 \, a g e\right )} \sqrt {x e + d}}{c^{3} d^{4} x + a c^{2} d^{2} x e^{2} + {\left (c^{3} d^{3} x^{2} + a c^{2} d^{3}\right )} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{\frac {3}{2}} \left (f + g x\right )}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.53, size = 127, normalized size = 0.85 \begin {gather*} \frac {2 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} g e^{\left (-1\right )}}{c^{2} d^{2}} + \frac {2 \, {\left (c d^{2} g + c d f e - 2 \, a g e^{2}\right )}}{\sqrt {-c d^{2} e + a e^{3}} c^{2} d^{2}} - \frac {2 \, {\left (c d f e - a g e^{2}\right )}}{\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.37, size = 118, normalized size = 0.79 \begin {gather*} \frac {\left (\frac {\left (4\,a\,e\,g-2\,c\,d\,f\right )\,\sqrt {d+e\,x}}{c^3\,d^3\,e}+\frac {2\,g\,x\,\sqrt {d+e\,x}}{c^2\,d^2\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{\frac {a}{c}+x^2+\frac {x\,\left (c^3\,d^4+a\,c^2\,d^2\,e^2\right )}{c^3\,d^3\,e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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