3.718 \(\int \frac{(d+e x)^m}{\sqrt{a+c x^2}} \, dx\)

Optimal. Leaf size=154 \[ \frac{(d+e x)^{m+1} \sqrt{1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}} \sqrt{1-\frac{d+e x}{\frac{\sqrt{-a} e}{\sqrt{c}}+d}} F_1\left (m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right )}{e (m+1) \sqrt{a+c x^2}} \]

[Out]

((d + e*x)^(1 + m)*Sqrt[1 - (d + e*x)/(d - (Sqrt[-a]*e)/Sqrt[c])]*Sqrt[1 - (d +
e*x)/(d + (Sqrt[-a]*e)/Sqrt[c])]*AppellF1[1 + m, 1/2, 1/2, 2 + m, (d + e*x)/(d -
(Sqrt[-a]*e)/Sqrt[c]), (d + e*x)/(d + (Sqrt[-a]*e)/Sqrt[c])])/(e*(1 + m)*Sqrt[a
+ c*x^2])

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Rubi [A]  time = 0.407257, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{(d+e x)^{m+1} \sqrt{1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}} \sqrt{1-\frac{d+e x}{\frac{\sqrt{-a} e}{\sqrt{c}}+d}} F_1\left (m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right )}{e (m+1) \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^m/Sqrt[a + c*x^2],x]

[Out]

((d + e*x)^(1 + m)*Sqrt[1 - (d + e*x)/(d - (Sqrt[-a]*e)/Sqrt[c])]*Sqrt[1 - (d +
e*x)/(d + (Sqrt[-a]*e)/Sqrt[c])]*AppellF1[1 + m, 1/2, 1/2, 2 + m, (d + e*x)/(d -
(Sqrt[-a]*e)/Sqrt[c]), (d + e*x)/(d + (Sqrt[-a]*e)/Sqrt[c])])/(e*(1 + m)*Sqrt[a
+ c*x^2])

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Rubi in Sympy [A]  time = 30.9284, size = 144, normalized size = 0.94 \[ \frac{\left (d + e x\right )^{m + 1} \sqrt{\frac{\sqrt{c} \left (- d - e x\right )}{\sqrt{c} d - e \sqrt{- a}} + 1} \sqrt{\frac{\sqrt{c} \left (- d - e x\right )}{\sqrt{c} d + e \sqrt{- a}} + 1} \operatorname{appellf_{1}}{\left (m + 1,\frac{1}{2},\frac{1}{2},m + 2,\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d - e \sqrt{- a}},\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d + e \sqrt{- a}} \right )}}{e \sqrt{a + c x^{2}} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**m/(c*x**2+a)**(1/2),x)

[Out]

(d + e*x)**(m + 1)*sqrt(sqrt(c)*(-d - e*x)/(sqrt(c)*d - e*sqrt(-a)) + 1)*sqrt(sq
rt(c)*(-d - e*x)/(sqrt(c)*d + e*sqrt(-a)) + 1)*appellf1(m + 1, 1/2, 1/2, m + 2,
sqrt(c)*(d + e*x)/(sqrt(c)*d - e*sqrt(-a)), sqrt(c)*(d + e*x)/(sqrt(c)*d + e*sqr
t(-a)))/(e*sqrt(a + c*x**2)*(m + 1))

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Mathematica [A]  time = 0.179803, size = 159, normalized size = 1.03 \[ \frac{(d+e x)^{m+1} \sqrt{\frac{e \left (\sqrt{-\frac{a}{c}}-x\right )}{e \sqrt{-\frac{a}{c}}+d}} \sqrt{\frac{e \left (\sqrt{-\frac{a}{c}}+x\right )}{e \sqrt{-\frac{a}{c}}-d}} F_1\left (m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{d+e x}{d-\sqrt{-\frac{a}{c}} e},\frac{d+e x}{d+\sqrt{-\frac{a}{c}} e}\right )}{e (m+1) \sqrt{a+c x^2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(d + e*x)^m/Sqrt[a + c*x^2],x]

[Out]

(Sqrt[(e*(Sqrt[-(a/c)] - x))/(d + Sqrt[-(a/c)]*e)]*Sqrt[(e*(Sqrt[-(a/c)] + x))/(
-d + Sqrt[-(a/c)]*e)]*(d + e*x)^(1 + m)*AppellF1[1 + m, 1/2, 1/2, 2 + m, (d + e*
x)/(d - Sqrt[-(a/c)]*e), (d + e*x)/(d + Sqrt[-(a/c)]*e)])/(e*(1 + m)*Sqrt[a + c*
x^2])

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \[ \int{ \left ( ex+d \right ) ^{m}{\frac{1}{\sqrt{c{x}^{2}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m/(c*x^2+a)^(1/2),x)

[Out]

int((e*x+d)^m/(c*x^2+a)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{m}}{\sqrt{c x^{2} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^m/sqrt(c*x^2 + a),x, algorithm="maxima")

[Out]

integrate((e*x + d)^m/sqrt(c*x^2 + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x + d\right )}^{m}}{\sqrt{c x^{2} + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^m/sqrt(c*x^2 + a),x, algorithm="fricas")

[Out]

integral((e*x + d)^m/sqrt(c*x^2 + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{m}}{\sqrt{a + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**m/(c*x**2+a)**(1/2),x)

[Out]

Integral((d + e*x)**m/sqrt(a + c*x**2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{m}}{\sqrt{c x^{2} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^m/sqrt(c*x^2 + a),x, algorithm="giac")

[Out]

integrate((e*x + d)^m/sqrt(c*x^2 + a), x)