### 3.119 $$\int (d x)^m (b x+c x^2)^{3/2} \, dx$$

Optimal. Leaf size=71 $-\frac{2 b (b+c x) \left (b x+c x^2\right )^{3/2} (d x)^m \left (-\frac{c x}{b}\right )^{-m-\frac{1}{2}} \, _2F_1\left (\frac{5}{2},-m-\frac{3}{2};\frac{7}{2};\frac{c x}{b}+1\right )}{5 c^2 x}$

[Out]

(-2*b*(-((c*x)/b))^(-1/2 - m)*(d*x)^m*(b + c*x)*(b*x + c*x^2)^(3/2)*Hypergeometric2F1[5/2, -3/2 - m, 7/2, 1 +
(c*x)/b])/(5*c^2*x)

________________________________________________________________________________________

Rubi [A]  time = 0.0294293, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.158, Rules used = {674, 67, 65} $-\frac{2 b (b+c x) \left (b x+c x^2\right )^{3/2} (d x)^m \left (-\frac{c x}{b}\right )^{-m-\frac{1}{2}} \, _2F_1\left (\frac{5}{2},-m-\frac{3}{2};\frac{7}{2};\frac{c x}{b}+1\right )}{5 c^2 x}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d*x)^m*(b*x + c*x^2)^(3/2),x]

[Out]

(-2*b*(-((c*x)/b))^(-1/2 - m)*(d*x)^m*(b + c*x)*(b*x + c*x^2)^(3/2)*Hypergeometric2F1[5/2, -3/2 - m, 7/2, 1 +
(c*x)/b])/(5*c^2*x)

Rule 674

Int[((e_.)*(x_))^(m_)*((b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[((e*x)^m*(b*x + c*x^2)^p)/(x^(m + p)
*(b + c*x)^p), Int[x^(m + p)*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, m}, x] &&  !IntegerQ[p]

Rule 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[((-((b*c)/d))^IntPart[m]*(b*x)^FracPart[m])/
(-((d*x)/c))^FracPart[m], Int[(-((d*x)/c))^m*(c + d*x)^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !GtQ[c, 0] &&  !GtQ[-(d/(b*c)), 0]

Rule 65

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)*Hypergeometric2F1[-m, n +
1, n + 2, 1 + (d*x)/c])/(d*(n + 1)*(-(d/(b*c)))^m), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Inte
gerQ[m] || GtQ[-(d/(b*c)), 0])

Rubi steps

\begin{align*} \int (d x)^m \left (b x+c x^2\right )^{3/2} \, dx &=\frac{\left (x^{-\frac{3}{2}-m} (d x)^m \left (b x+c x^2\right )^{3/2}\right ) \int x^{\frac{3}{2}+m} (b+c x)^{3/2} \, dx}{(b+c x)^{3/2}}\\ &=-\frac{\left (b \left (-\frac{c x}{b}\right )^{-\frac{1}{2}-m} (d x)^m \left (b x+c x^2\right )^{3/2}\right ) \int \left (-\frac{c x}{b}\right )^{\frac{3}{2}+m} (b+c x)^{3/2} \, dx}{c x (b+c x)^{3/2}}\\ &=-\frac{2 b \left (-\frac{c x}{b}\right )^{-\frac{1}{2}-m} (d x)^m (b+c x) \left (b x+c x^2\right )^{3/2} \, _2F_1\left (\frac{5}{2},-\frac{3}{2}-m;\frac{7}{2};1+\frac{c x}{b}\right )}{5 c^2 x}\\ \end{align*}

Mathematica [A]  time = 0.116832, size = 60, normalized size = 0.85 $-\frac{2 (x (b+c x))^{5/2} (d x)^m \left (-\frac{c x}{b}\right )^{-m-\frac{5}{2}} \, _2F_1\left (\frac{5}{2},-m-\frac{3}{2};\frac{7}{2};\frac{c x}{b}+1\right )}{5 b}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*(-((c*x)/b))^(-5/2 - m)*(d*x)^m*(x*(b + c*x))^(5/2)*Hypergeometric2F1[5/2, -3/2 - m, 7/2, 1 + (c*x)/b])/(5
*b)

________________________________________________________________________________________

Maple [F]  time = 0.403, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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