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

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

[Out]

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

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Rubi [A]  time = 0.408205, 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 \[ \frac{\sqrt{a+c x^2} (d+e x)^{m+1} F_1\left (m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right )}{e (m+1) \sqrt{1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}} \sqrt{1-\frac{d+e x}{\frac{\sqrt{-a} e}{\sqrt{c}}+d}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 30.6796, size = 148, normalized size = 0.96 \[ \frac{\sqrt{a + c x^{2}} \left (d + e x\right )^{m + 1} \operatorname{appellf_{1}}{\left (m + 1,- \frac{1}{2},- \frac{1}{2},m + 2,\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d - e \sqrt{- a}},\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d + e \sqrt{- a}} \right )}}{e \left (m + 1\right ) \sqrt{\frac{\sqrt{c} \left (- d - e x\right )}{\sqrt{c} d - e \sqrt{- a}} + 1} \sqrt{\frac{\sqrt{c} \left (- d - e x\right )}{\sqrt{c} d + e \sqrt{- a}} + 1}} \]

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]

sqrt(a + c*x**2)*(d + e*x)**(m + 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*sqrt(-a)))/(
e*(m + 1)*sqrt(sqrt(c)*(-d - e*x)/(sqrt(c)*d - e*sqrt(-a)) + 1)*sqrt(sqrt(c)*(-d
- e*x)/(sqrt(c)*d + e*sqrt(-a)) + 1))

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Mathematica [A]  time = 0.179224, size = 159, normalized size = 1.03 \[ \frac{\sqrt{a+c x^2} (d+e x)^{m+1} F_1\left (m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{d+e x}{d-\sqrt{-\frac{a}{c}} e},\frac{d+e x}{d+\sqrt{-\frac{a}{c}} e}\right )}{e (m+1) \sqrt{\frac{e \left (\sqrt{-\frac{a}{c}}-x\right )}{e \sqrt{-\frac{a}{c}}+d}} \sqrt{\frac{e \left (\sqrt{-\frac{a}{c}}+x\right )}{e \sqrt{-\frac{a}{c}}-d}}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{m}\sqrt{c{x}^{2}+a}\, dx \]

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. \[ \int \sqrt{c x^{2} + a}{\left (e x + d\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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(sqrt(a + c*x**2)*(d + e*x)**m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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