Optimal. Leaf size=209 \[ -\frac {4 \left (c d^2-a e^2\right ) \left (4 a e^2 g-c d (5 e f-d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 c^3 d^3 e \sqrt {d+e x}}-\frac {2 \left (4 a e^2 g-c d (5 e f-d g)\right ) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 c^2 d^2 e}+\frac {2 g (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 c d e} \]
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Rubi [A]
time = 0.13, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {808, 670, 662}
\begin {gather*} -\frac {4 \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} \left (4 a e^2 g-c d (5 e f-d g)\right )}{15 c^3 d^3 e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (4 a e^2 g-c d (5 e f-d g)\right )}{15 c^2 d^2 e}+\frac {2 g (d+e x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d e} \end {gather*}
Antiderivative was successfully verified.
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Rule 662
Rule 670
Rule 808
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2} (f+g x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {2 g (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 c d e}+\frac {1}{5} \left (5 f-\frac {d g}{e}-\frac {4 a e g}{c d}\right ) \int \frac {(d+e x)^{3/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\\ &=-\frac {2 \left (4 a e^2 g-c d (5 e f-d g)\right ) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 c^2 d^2 e}+\frac {2 g (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 c d e}+\frac {\left (2 \left (d^2-\frac {a e^2}{c}\right ) \left (5 f-\frac {d g}{e}-\frac {4 a e g}{c d}\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}{15 d}\\ &=-\frac {4 \left (c d^2-a e^2\right ) \left (4 a e^2 g-c d (5 e f-d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 c^3 d^3 e \sqrt {d+e x}}-\frac {2 \left (4 a e^2 g-c d (5 e f-d g)\right ) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 c^2 d^2 e}+\frac {2 g (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 c d e}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 96, normalized size = 0.46 \begin {gather*} \frac {2 \sqrt {(a e+c d x) (d+e x)} \left (8 a^2 e^3 g-2 a c d e (5 e f+5 d g+2 e g x)+c^2 d^2 (5 d (3 f+g x)+e x (5 f+3 g x))\right )}{15 c^3 d^3 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 113, normalized size = 0.54
method | result | size |
default | \(\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (3 e g \,x^{2} c^{2} d^{2}-4 a c d \,e^{2} g x +5 c^{2} d^{3} g x +5 c^{2} d^{2} e f x +8 a^{2} e^{3} g -10 a c \,d^{2} e g -10 a c d \,e^{2} f +15 d^{3} f \,c^{2}\right )}{15 \sqrt {e x +d}\, c^{3} d^{3}}\) | \(113\) |
gosper | \(\frac {2 \left (c d x +a e \right ) \left (3 e g \,x^{2} c^{2} d^{2}-4 a c d \,e^{2} g x +5 c^{2} d^{3} g x +5 c^{2} d^{2} e f x +8 a^{2} e^{3} g -10 a c \,d^{2} e g -10 a c d \,e^{2} f +15 d^{3} f \,c^{2}\right ) \sqrt {e x +d}}{15 c^{3} d^{3} \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}\) | \(131\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 168, normalized size = 0.80 \begin {gather*} \frac {2 \, {\left (c^{2} d^{2} x^{2} e + 3 \, a c d^{2} e - 2 \, a^{2} e^{3} + {\left (3 \, c^{2} d^{3} - a c d e^{2}\right )} x\right )} f}{3 \, \sqrt {c d x + a e} c^{2} d^{2}} + \frac {2 \, {\left (3 \, c^{3} d^{3} x^{3} e - 10 \, a^{2} c d^{2} e^{2} + 8 \, a^{3} e^{4} + {\left (5 \, c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} x^{2} - {\left (5 \, a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} x\right )} g}{15 \, \sqrt {c d x + a e} c^{3} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.93, size = 137, normalized size = 0.66 \begin {gather*} \frac {2 \, {\left (5 \, c^{2} d^{3} g x + 15 \, c^{2} d^{3} f + 8 \, a^{2} g e^{3} - 2 \, {\left (2 \, a c d g x + 5 \, a c d f\right )} e^{2} + {\left (3 \, c^{2} d^{2} g x^{2} + 5 \, c^{2} d^{2} f x - 10 \, a c d^{2} g\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{15 \, {\left (c^{3} d^{3} x e + c^{3} d^{4}\right )}} \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 )}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.25, size = 342, normalized size = 1.64 \begin {gather*} \frac {2 \, {\left (c^{2} d^{3} f - a c d^{2} g e - a c d f e^{2} + a^{2} g e^{3}\right )} \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} e^{\left (-1\right )}}{c^{3} d^{3}} + \frac {4 \, {\left (\sqrt {-c d^{2} e + a e^{3}} c^{2} d^{4} g - 5 \, \sqrt {-c d^{2} e + a e^{3}} c^{2} d^{3} f e + 3 \, \sqrt {-c d^{2} e + a e^{3}} a c d^{2} g e^{2} + 5 \, \sqrt {-c d^{2} e + a e^{3}} a c d f e^{3} - 4 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} g e^{4}\right )} e^{\left (-2\right )}}{15 \, c^{3} d^{3}} + \frac {2 \, {\left (5 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c d^{2} g e + 5 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c d f e^{2} - 10 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a g e^{3} + 3 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} g\right )} e^{\left (-4\right )}}{15 \, c^{3} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.48, size = 152, normalized size = 0.73 \begin {gather*} \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {\sqrt {d+e\,x}\,\left (16\,g\,a^2\,e^3-20\,g\,a\,c\,d^2\,e-20\,f\,a\,c\,d\,e^2+30\,f\,c^2\,d^3\right )}{15\,c^3\,d^3\,e}+\frac {2\,g\,x^2\,\sqrt {d+e\,x}}{5\,c\,d}+\frac {2\,x\,\sqrt {d+e\,x}\,\left (5\,c\,g\,d^2+5\,c\,f\,d\,e-4\,a\,g\,e^2\right )}{15\,c^2\,d^2\,e}\right )}{x+\frac {d}{e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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