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

Optimal. Leaf size=105 \[ \frac{\left (b x+c x^2\right )^{3/2} (d+e x)^{m+1} F_1\left (m+1;-\frac{3}{2},-\frac{3}{2};m+2;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right )}{e (m+1) \left (-\frac{e x}{d}\right )^{3/2} \left (1-\frac{c (d+e x)}{c d-b e}\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.252811, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{\left (b x+c x^2\right )^{3/2} (d+e x)^{m+1} F_1\left (m+1;-\frac{3}{2},-\frac{3}{2};m+2;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right )}{e (m+1) \left (-\frac{e x}{d}\right )^{3/2} \left (1-\frac{c (d+e x)}{c d-b e}\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.8215, size = 85, normalized size = 0.81 \[ \frac{\left (d + e x\right )^{m + 1} \left (b x + c x^{2}\right )^{\frac{3}{2}} \operatorname{appellf_{1}}{\left (m + 1,- \frac{3}{2},- \frac{3}{2},m + 2,\frac{d + e x}{d},\frac{c \left (- d - e x\right )}{b e - c d} \right )}}{e \left (- \frac{e x}{d}\right )^{\frac{3}{2}} \left (m + 1\right ) \left (\frac{c \left (d + e x\right )}{b e - c d} + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.941768, size = 289, normalized size = 2.75 \[ \frac{2}{35} b d x^2 \sqrt{x (b+c x)} (d+e x)^m \left (\frac{49 b F_1\left (\frac{5}{2};-\frac{1}{2},-m;\frac{7}{2};-\frac{c x}{b},-\frac{e x}{d}\right )}{7 b d F_1\left (\frac{5}{2};-\frac{1}{2},-m;\frac{7}{2};-\frac{c x}{b},-\frac{e x}{d}\right )+2 b e m x F_1\left (\frac{7}{2};-\frac{1}{2},1-m;\frac{9}{2};-\frac{c x}{b},-\frac{e x}{d}\right )+c d x F_1\left (\frac{7}{2};\frac{1}{2},-m;\frac{9}{2};-\frac{c x}{b},-\frac{e x}{d}\right )}+\frac{45 c x F_1\left (\frac{7}{2};-\frac{1}{2},-m;\frac{9}{2};-\frac{c x}{b},-\frac{e x}{d}\right )}{9 b d F_1\left (\frac{7}{2};-\frac{1}{2},-m;\frac{9}{2};-\frac{c x}{b},-\frac{e x}{d}\right )+2 b e m x F_1\left (\frac{9}{2};-\frac{1}{2},1-m;\frac{11}{2};-\frac{c x}{b},-\frac{e x}{d}\right )+c d x F_1\left (\frac{9}{2};\frac{1}{2},-m;\frac{11}{2};-\frac{c x}{b},-\frac{e x}{d}\right )}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*b*d*x^2*Sqrt[x*(b + c*x)]*(d + e*x)^m*((49*b*AppellF1[5/2, -1/2, -m, 7/2, -((
c*x)/b), -((e*x)/d)])/(7*b*d*AppellF1[5/2, -1/2, -m, 7/2, -((c*x)/b), -((e*x)/d)
] + 2*b*e*m*x*AppellF1[7/2, -1/2, 1 - m, 9/2, -((c*x)/b), -((e*x)/d)] + c*d*x*Ap
pellF1[7/2, 1/2, -m, 9/2, -((c*x)/b), -((e*x)/d)]) + (45*c*x*AppellF1[7/2, -1/2,
-m, 9/2, -((c*x)/b), -((e*x)/d)])/(9*b*d*AppellF1[7/2, -1/2, -m, 9/2, -((c*x)/b
), -((e*x)/d)] + 2*b*e*m*x*AppellF1[9/2, -1/2, 1 - m, 11/2, -((c*x)/b), -((e*x)/
d)] + c*d*x*AppellF1[9/2, 1/2, -m, 11/2, -((c*x)/b), -((e*x)/d)])))/35

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{m} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*x^2 + b*x)^(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+b*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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