3.716 \(\int (d+e x)^m \left (a+c x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=154 \[ \frac{\left (a+c x^2\right )^{3/2} (d+e x)^{m+1} F_1\left (m+1;-\frac{3}{2},-\frac{3}{2};m+2;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right )}{e (m+1) \left (1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}\right )^{3/2} \left (1-\frac{d+e x}{\frac{\sqrt{-a} e}{\sqrt{c}}+d}\right )^{3/2}} \]

[Out]

((d + e*x)^(1 + m)*(a + c*x^2)^(3/2)*AppellF1[1 + m, -3/2, -3/2, 2 + m, (d + e*x
)/(d - (Sqrt[-a]*e)/Sqrt[c]), (d + e*x)/(d + (Sqrt[-a]*e)/Sqrt[c])])/(e*(1 + m)*
(1 - (d + e*x)/(d - (Sqrt[-a]*e)/Sqrt[c]))^(3/2)*(1 - (d + e*x)/(d + (Sqrt[-a]*e
)/Sqrt[c]))^(3/2))

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Rubi [A]  time = 0.526024, 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 \[ \frac{\left (a+c x^2\right )^{3/2} (d+e x)^{m+1} F_1\left (m+1;-\frac{3}{2},-\frac{3}{2};m+2;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right )}{e (m+1) \left (1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}\right )^{3/2} \left (1-\frac{d+e x}{\frac{\sqrt{-a} e}{\sqrt{c}}+d}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^m*(a + c*x^2)^(3/2),x]

[Out]

((d + e*x)^(1 + m)*(a + c*x^2)^(3/2)*AppellF1[1 + m, -3/2, -3/2, 2 + m, (d + e*x
)/(d - (Sqrt[-a]*e)/Sqrt[c]), (d + e*x)/(d + (Sqrt[-a]*e)/Sqrt[c])])/(e*(1 + m)*
(1 - (d + e*x)/(d - (Sqrt[-a]*e)/Sqrt[c]))^(3/2)*(1 - (d + e*x)/(d + (Sqrt[-a]*e
)/Sqrt[c]))^(3/2))

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Rubi in Sympy [A]  time = 30.6979, size = 148, normalized size = 0.96 \[ \frac{\left (a + c x^{2}\right )^{\frac{3}{2}} \left (d + e x\right )^{m + 1} \operatorname{appellf_{1}}{\left (m + 1,- \frac{3}{2},- \frac{3}{2},m + 2,\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d - e \sqrt{- a}},\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d + e \sqrt{- a}} \right )}}{e \left (m + 1\right ) \left (\frac{\sqrt{c} \left (- d - e x\right )}{\sqrt{c} d - e \sqrt{- a}} + 1\right )^{\frac{3}{2}} \left (\frac{\sqrt{c} \left (- d - e x\right )}{\sqrt{c} d + e \sqrt{- a}} + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**m*(c*x**2+a)**(3/2),x)

[Out]

(a + c*x**2)**(3/2)*(d + e*x)**(m + 1)*appellf1(m + 1, -3/2, -3/2, 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(c)*(-d - e*x)/(sqrt(c)*d - e*sqrt(-a)) + 1)**(3/2)*(sqrt(c)*(
-d - e*x)/(sqrt(c)*d + e*sqrt(-a)) + 1)**(3/2))

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Mathematica [A]  time = 0.118075, size = 0, normalized size = 0. \[ \int (d+e x)^m \left (a+c x^2\right )^{3/2} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(d + e*x)^m*(a + c*x^2)^(3/2),x]

[Out]

Integrate[(d + e*x)^m*(a + c*x^2)^(3/2), x]

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Maple [F]  time = 0.048, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{m} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m*(c*x^2+a)^(3/2),x)

[Out]

int((e*x+d)^m*(c*x^2+a)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + a\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)*(e*x + d)^m,x, algorithm="maxima")

[Out]

integrate((c*x^2 + a)^(3/2)*(e*x + d)^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{2} + a\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)*(e*x + d)^m,x, algorithm="fricas")

[Out]

integral((c*x^2 + a)^(3/2)*(e*x + d)^m, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**m*(c*x**2+a)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + a\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)*(e*x + d)^m,x, algorithm="giac")

[Out]

integrate((c*x^2 + a)^(3/2)*(e*x + d)^m, x)