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

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

[Out]

((-((e*x)/d))^(3/2)*(d + e*x)^(1 + m)*(1 - (c*(d + e*x))/(c*d - b*e))^(3/2)*Appe
llF1[1 + m, 3/2, 3/2, 2 + m, (d + e*x)/d, (c*(d + e*x))/(c*d - b*e)])/(e*(1 + m)
*(b*x + c*x^2)^(3/2))

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Rubi [A]  time = 0.24376, 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 \[ \frac{\left (-\frac{e x}{d}\right )^{3/2} (d+e x)^{m+1} \left (1-\frac{c (d+e x)}{c d-b e}\right )^{3/2} F_1\left (m+1;\frac{3}{2},\frac{3}{2};m+2;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right )}{e (m+1) \left (b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((-((e*x)/d))^(3/2)*(d + e*x)^(1 + m)*(1 - (c*(d + e*x))/(c*d - b*e))^(3/2)*Appe
llF1[1 + m, 3/2, 3/2, 2 + m, (d + e*x)/d, (c*(d + e*x))/(c*d - b*e)])/(e*(1 + m)
*(b*x + c*x^2)^(3/2))

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Rubi in Sympy [A]  time = 20.7212, size = 82, normalized size = 0.78 \[ \frac{\left (- \frac{e x}{d}\right )^{\frac{3}{2}} \left (d + e x\right )^{m + 1} \left (\frac{c \left (d + e x\right )}{b e - c d} + 1\right )^{\frac{3}{2}} \operatorname{appellf_{1}}{\left (m + 1,\frac{3}{2},\frac{3}{2},m + 2,\frac{d + e x}{d},\frac{c \left (- d - e x\right )}{b e - c d} \right )}}{e \left (m + 1\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}} \]

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]

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

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Mathematica [B]  time = 1.45904, size = 433, normalized size = 4.12 \[ \frac{2 d x (d+e x)^m \left (-\frac{(b+c x)^2 F_1\left (-\frac{1}{2};-\frac{1}{2},-m;\frac{1}{2};-\frac{c x}{b},-\frac{e x}{d}\right )}{b \left (b d F_1\left (-\frac{1}{2};-\frac{1}{2},-m;\frac{1}{2};-\frac{c x}{b},-\frac{e x}{d}\right )+2 b e m x F_1\left (\frac{1}{2};-\frac{1}{2},1-m;\frac{3}{2};-\frac{c x}{b},-\frac{e x}{d}\right )+c d x F_1\left (\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};-\frac{c x}{b},-\frac{e x}{d}\right )\right )}-\frac{3 c x (b+c x) F_1\left (\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};-\frac{c x}{b},-\frac{e x}{d}\right )}{b \left (3 b d F_1\left (\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};-\frac{c x}{b},-\frac{e x}{d}\right )+2 b e m x F_1\left (\frac{3}{2};\frac{1}{2},1-m;\frac{5}{2};-\frac{c x}{b},-\frac{e x}{d}\right )-c d x F_1\left (\frac{3}{2};\frac{3}{2},-m;\frac{5}{2};-\frac{c x}{b},-\frac{e x}{d}\right )\right )}-\frac{3 c x F_1\left (\frac{1}{2};\frac{3}{2},-m;\frac{3}{2};-\frac{c x}{b},-\frac{e x}{d}\right )}{3 b d F_1\left (\frac{1}{2};\frac{3}{2},-m;\frac{3}{2};-\frac{c x}{b},-\frac{e x}{d}\right )+2 b e m x F_1\left (\frac{3}{2};\frac{3}{2},1-m;\frac{5}{2};-\frac{c x}{b},-\frac{e x}{d}\right )-3 c d x F_1\left (\frac{3}{2};\frac{5}{2},-m;\frac{5}{2};-\frac{c x}{b},-\frac{e x}{d}\right )}\right )}{(x (b+c x))^{3/2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*d*x*(d + e*x)^m*(-(((b + c*x)^2*AppellF1[-1/2, -1/2, -m, 1/2, -((c*x)/b), -((
e*x)/d)])/(b*(b*d*AppellF1[-1/2, -1/2, -m, 1/2, -((c*x)/b), -((e*x)/d)] + 2*b*e*
m*x*AppellF1[1/2, -1/2, 1 - m, 3/2, -((c*x)/b), -((e*x)/d)] + c*d*x*AppellF1[1/2
, 1/2, -m, 3/2, -((c*x)/b), -((e*x)/d)]))) - (3*c*x*(b + c*x)*AppellF1[1/2, 1/2,
-m, 3/2, -((c*x)/b), -((e*x)/d)])/(b*(3*b*d*AppellF1[1/2, 1/2, -m, 3/2, -((c*x)
/b), -((e*x)/d)] + 2*b*e*m*x*AppellF1[3/2, 1/2, 1 - m, 5/2, -((c*x)/b), -((e*x)/
d)] - c*d*x*AppellF1[3/2, 3/2, -m, 5/2, -((c*x)/b), -((e*x)/d)])) - (3*c*x*Appel
lF1[1/2, 3/2, -m, 3/2, -((c*x)/b), -((e*x)/d)])/(3*b*d*AppellF1[1/2, 3/2, -m, 3/
2, -((c*x)/b), -((e*x)/d)] + 2*b*e*m*x*AppellF1[3/2, 3/2, 1 - m, 5/2, -((c*x)/b)
, -((e*x)/d)] - 3*c*d*x*AppellF1[3/2, 5/2, -m, 5/2, -((c*x)/b), -((e*x)/d)])))/(
x*(b + c*x))^(3/2)

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Maple [F]  time = 0.049, 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. \[ \int \frac{{\left (e x + d\right )}^{m}}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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