3.207 \(\int (d+e x)^m \log (c (a+b x^2)^p) \, dx\)

Optimal. Leaf size=205 \[ \frac{(d+e x)^{m+1} \log \left (c \left (a+b x^2\right )^p\right )}{e (m+1)}+\frac{\sqrt{b} p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac{\sqrt{b} (d+e x)}{\sqrt{b} d-\sqrt{-a} e}\right )}{e (m+1) (m+2) \left (\sqrt{b} d-\sqrt{-a} e\right )}+\frac{\sqrt{b} p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac{\sqrt{b} (d+e x)}{\sqrt{b} d+\sqrt{-a} e}\right )}{e (m+1) (m+2) \left (\sqrt{-a} e+\sqrt{b} d\right )} \]

[Out]

(Sqrt[b]*p*(d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d - Sqrt[-a]*e)])
/(e*(Sqrt[b]*d - Sqrt[-a]*e)*(1 + m)*(2 + m)) + (Sqrt[b]*p*(d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m
, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d + Sqrt[-a]*e)])/(e*(Sqrt[b]*d + Sqrt[-a]*e)*(1 + m)*(2 + m)) + ((d + e*x)^(1
+ m)*Log[c*(a + b*x^2)^p])/(e*(1 + m))

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Rubi [A]  time = 0.247979, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2463, 831, 68} \[ \frac{(d+e x)^{m+1} \log \left (c \left (a+b x^2\right )^p\right )}{e (m+1)}+\frac{\sqrt{b} p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac{\sqrt{b} (d+e x)}{\sqrt{b} d-\sqrt{-a} e}\right )}{e (m+1) (m+2) \left (\sqrt{b} d-\sqrt{-a} e\right )}+\frac{\sqrt{b} p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac{\sqrt{b} (d+e x)}{\sqrt{b} d+\sqrt{-a} e}\right )}{e (m+1) (m+2) \left (\sqrt{-a} e+\sqrt{b} d\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b]*p*(d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d - Sqrt[-a]*e)])
/(e*(Sqrt[b]*d - Sqrt[-a]*e)*(1 + m)*(2 + m)) + (Sqrt[b]*p*(d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m
, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d + Sqrt[-a]*e)])/(e*(Sqrt[b]*d + Sqrt[-a]*e)*(1 + m)*(2 + m)) + ((d + e*x)^(1
+ m)*Log[c*(a + b*x^2)^p])/(e*(1 + m))

Rule 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.) + (g_.)*(x_))^(r_.), x_Symbol] :> Simp[((
f + g*x)^(r + 1)*(a + b*Log[c*(d + e*x^n)^p]))/(g*(r + 1)), x] - Dist[(b*e*n*p)/(g*(r + 1)), Int[(x^(n - 1)*(f
+ g*x)^(r + 1))/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, r}, x] && (IGtQ[r, 0] || RationalQ[n
]) && NeQ[r, -1]

Rule 831

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x)^m, (f + g*x)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !Ration
alQ[m]

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rubi steps

\begin{align*} \int (d+e x)^m \log \left (c \left (a+b x^2\right )^p\right ) \, dx &=\frac{(d+e x)^{1+m} \log \left (c \left (a+b x^2\right )^p\right )}{e (1+m)}-\frac{(2 b p) \int \frac{x (d+e x)^{1+m}}{a+b x^2} \, dx}{e (1+m)}\\ &=\frac{(d+e x)^{1+m} \log \left (c \left (a+b x^2\right )^p\right )}{e (1+m)}-\frac{(2 b p) \int \left (-\frac{(d+e x)^{1+m}}{2 \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{(d+e x)^{1+m}}{2 \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )}\right ) \, dx}{e (1+m)}\\ &=\frac{(d+e x)^{1+m} \log \left (c \left (a+b x^2\right )^p\right )}{e (1+m)}+\frac{\left (\sqrt{b} p\right ) \int \frac{(d+e x)^{1+m}}{\sqrt{-a}-\sqrt{b} x} \, dx}{e (1+m)}-\frac{\left (\sqrt{b} p\right ) \int \frac{(d+e x)^{1+m}}{\sqrt{-a}+\sqrt{b} x} \, dx}{e (1+m)}\\ &=\frac{\sqrt{b} p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac{\sqrt{b} (d+e x)}{\sqrt{b} d-\sqrt{-a} e}\right )}{e \left (\sqrt{b} d-\sqrt{-a} e\right ) (1+m) (2+m)}+\frac{\sqrt{b} p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac{\sqrt{b} (d+e x)}{\sqrt{b} d+\sqrt{-a} e}\right )}{e \left (\sqrt{b} d+\sqrt{-a} e\right ) (1+m) (2+m)}+\frac{(d+e x)^{1+m} \log \left (c \left (a+b x^2\right )^p\right )}{e (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.276107, size = 176, normalized size = 0.86 \[ \frac{(d+e x)^{m+1} \left (\log \left (c \left (a+b x^2\right )^p\right )+\frac{\sqrt{b} p (d+e x) \left (\left (\sqrt{-a} e+\sqrt{b} d\right ) \, _2F_1\left (1,m+2;m+3;\frac{\sqrt{b} (d+e x)}{\sqrt{b} d-\sqrt{-a} e}\right )+\left (\sqrt{b} d-\sqrt{-a} e\right ) \, _2F_1\left (1,m+2;m+3;\frac{\sqrt{b} (d+e x)}{\sqrt{b} d+\sqrt{-a} e}\right )\right )}{(m+2) \left (a e^2+b d^2\right )}\right )}{e (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(1 + m)*((Sqrt[b]*p*(d + e*x)*((Sqrt[b]*d + Sqrt[-a]*e)*Hypergeometric2F1[1, 2 + m, 3 + m, (Sqrt[b]
*(d + e*x))/(Sqrt[b]*d - Sqrt[-a]*e)] + (Sqrt[b]*d - Sqrt[-a]*e)*Hypergeometric2F1[1, 2 + m, 3 + m, (Sqrt[b]*(
d + e*x))/(Sqrt[b]*d + Sqrt[-a]*e)]))/((b*d^2 + a*e^2)*(2 + m)) + Log[c*(a + b*x^2)^p]))/(e*(1 + m))

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Maple [F]  time = 1.411, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{m}\ln \left ( c \left ( b{x}^{2}+a \right ) ^{p} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m*ln(c*(b*x^2+a)^p),x)

[Out]

int((e*x+d)^m*ln(c*(b*x^2+a)^p),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((e*x + d)^m*log((b*x^2 + a)^p*c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**m*ln(c*(b*x**2+a)**p),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x + d)^m*log((b*x^2 + a)^p*c), x)