3.2570 \(\int (d+e x)^{3/2} (a+b x+c x^2)^p \, dx\)

Optimal. Leaf size=185 \[ \frac {2 (d+e x)^{5/2} \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}\right )^{-p} F_1\left (\frac {5}{2};-p,-p;\frac {7}{2};\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{5 e} \]

[Out]

2/5*(e*x+d)^(5/2)*(c*x^2+b*x+a)^p*AppellF1(5/2,-p,-p,7/2,2*c*(e*x+d)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2))),2*c*(e*x
+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))/e/((1-2*c*(e*x+d)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2))))^p)/((1-2*c*(e*x+d)/(
2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^p)

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Rubi [A]  time = 0.15, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {759, 133} \[ \frac {2 (d+e x)^{5/2} \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}\right )^{-p} F_1\left (\frac {5}{2};-p,-p;\frac {7}{2};\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{5 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(a + b*x + c*x^2)^p*AppellF1[5/2, -p, -p, 7/2, (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*a
*c])*e), (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(5*e*(1 - (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^
2 - 4*a*c])*e))^p*(1 - (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e))^p)

Rule 133

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(c^n*e^p*(b*x)^(m +
1)*AppellF1[m + 1, -n, -p, m + 2, -((d*x)/c), -((f*x)/e)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rule 759

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c,
2]}, Dist[(a + b*x + c*x^2)^p/(e*(1 - (d + e*x)/(d - (e*(b - q))/(2*c)))^p*(1 - (d + e*x)/(d - (e*(b + q))/(2
*c)))^p), Subst[Int[x^m*Simp[1 - x/(d - (e*(b - q))/(2*c)), x]^p*Simp[1 - x/(d - (e*(b + q))/(2*c)), x]^p, x],
x, d + e*x], x]] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &
& NeQ[2*c*d - b*e, 0] &&  !IntegerQ[p]

Rubi steps

\begin {align*} \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^p \, dx &=\frac {\left (\left (a+b x+c x^2\right )^p \left (1-\frac {d+e x}{d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c}}\right )^{-p} \left (1-\frac {d+e x}{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}}\right )^{-p}\right ) \operatorname {Subst}\left (\int x^{3/2} \left (1-\frac {2 c x}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^p \left (1-\frac {2 c x}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^p \, dx,x,d+e x\right )}{e}\\ &=\frac {2 (d+e x)^{5/2} \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^{-p} F_1\left (\frac {5}{2};-p,-p;\frac {7}{2};\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{5 e}\\ \end {align*}

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Mathematica [A]  time = 1.40, size = 239, normalized size = 1.29 \[ \frac {2 (d+e x)^{5/2} (a+x (b+c x))^p \left (\frac {e \left (e \sqrt {\frac {b^2-4 a c}{e^2}}-b-2 c x\right )}{e \left (e \sqrt {\frac {b^2-4 a c}{e^2}}-b\right )+2 c d}\right )^{-p} \left (\frac {e \left (e \sqrt {\frac {b^2-4 a c}{e^2}}+b+2 c x\right )}{e \left (e \sqrt {\frac {b^2-4 a c}{e^2}}+b\right )-2 c d}\right )^{-p} F_1\left (\frac {5}{2};-p,-p;\frac {7}{2};\frac {2 c (d+e x)}{2 c d-e \left (b+\sqrt {\frac {b^2-4 a c}{e^2}} e\right )},\frac {2 c (d+e x)}{2 c d+e \left (\sqrt {\frac {b^2-4 a c}{e^2}} e-b\right )}\right )}{5 e} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(a + x*(b + c*x))^p*AppellF1[5/2, -p, -p, 7/2, (2*c*(d + e*x))/(2*c*d - e*(b + Sqrt[(b^2 -
4*a*c)/e^2]*e)), (2*c*(d + e*x))/(2*c*d + e*(-b + Sqrt[(b^2 - 4*a*c)/e^2]*e))])/(5*e*((e*(-b + Sqrt[(b^2 - 4*a
*c)/e^2]*e - 2*c*x))/(2*c*d + e*(-b + Sqrt[(b^2 - 4*a*c)/e^2]*e)))^p*((e*(b + Sqrt[(b^2 - 4*a*c)/e^2]*e + 2*c*
x))/(-2*c*d + e*(b + Sqrt[(b^2 - 4*a*c)/e^2]*e)))^p)

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fricas [F]  time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e x + d\right )}^{\frac {3}{2}} {\left (c x^{2} + b x + a\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(c*x^2+b*x+a)^p,x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(c*x^2+b*x+a)^p,x, algorithm="giac")

[Out]

sage0*x

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maple [F]  time = 1.57, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{\frac {3}{2}} \left (c \,x^{2}+b x +a \right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x + d\right )}^{\frac {3}{2}} {\left (c x^{2} + b x + a\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(c*x^2+b*x+a)^p,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d+e\,x\right )}^{3/2}\,{\left (c\,x^2+b\,x+a\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)*(c*x**2+b*x+a)**p,x)

[Out]

Timed out

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